Deutsches Zentrum fur¤ Luft- und Raumfahrt e.V. · Pablo Neruda ﬁ ...It’s never over ... Holzner for his comments on the chapter on ... Sonia, Silvia, Antonio, Millah, Zso,·

Deutsches Zentrumfur Luft- und Raumfahrt e.V.

Forschungsbericht 2004-05

Comparison of polarimetric methodsin image classification andSAR interferometry applications

Vito Alberga

Institut fur Hochfrequenztechnikund RadarsystemeOberpfaffenhofen

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Comparison of polarimetric methods inimage classification and

SAR interferometry applications

Zur Erlangung des akademischen Grades einesDoktoringenieurs (Dr.-Ing.)

von der Fakultat fur Elektrotechnik und Informationstechnikder Technischen Universitat Chemnitz

genehmigte Dissertation von

Dott. Vito Alberga

Referent: Prof. Dr.-Ing. G. WanielikKorreferent: Prof. Dr. M. ChandraKorreferent: Dr. E. Krogager

Tag der Einreichung: 29. Juli 2003Tag der mundlichen Prufung: 23. Januar 2004

Abstract

In this thesis, an analysis of the various parameters derivable from polarimetric SAR measurements isreported. The theory related to polarimetry and the state of the art of its application to remote sensingof the Earth by means of SAR systems are thoroughly discussed.

The experimental part of this work pursues the task of analyzing all the relevant polarimetric param-eters. In the first part of the thesis, a systematic study of the different ways of examining polarimetricdata has been performed. The main aim was to evaluate possible differences among the various polari-metric observables and the amount and usefulness of the information they contain. In this context, theseobservables have been compared by means of the accuracy estimates resulting from classification testsof real polarimetric SAR data. In the analysis proposed here, such accuracy estimates act as an objectivemeasure of the “utility” of the observables.

In the second part, some of the aforementioned polarimetric observables have been used for inter-ferometric applications. The main objective was to determine if a characterization of volume scattering,one of the terms affecting the interferometric coherence, is possible. Once again a comparison of theselected parameters has been done in terms of their capability to reduce the effects of volume scatteringin interferometric coherence images.

Since this work is intended as a general survey of polarimetric observables, completeness has beenan important goal, which the author hopes to have achieved. The twofold view of the investigationsreported here, oriented both to classification and interferometry, contributes to a comprehensive under-standing of the parameters under study.

Keywords: Radar imaging, synthetic aperture radar (SAR), SAR polarimetry, SAR interferometry, polarimet-ric SAR interferometry, image classification.

Kurzfassung

In dieser Arbeit wird uber die Analyse von unterschiedlichen Parametern, ermittelt vom polarimetri-schen SAR, berichtet. Die Theorie der Polarimetrie und der Stand der Technik in der Radarfernerkun-dung der Erdoberflache, die auf SAR-Systemen beruhe, ist grundlich dargestellt.

Der experimentelle Anteil dieser Arbeit beinhaltet die Analyse von allen relevanten polarimetrischenParametern. Der erste Teil ist eine systematische Untersuchung von polarimetrischen Daten, wobeiunterschiedliche Wege analysiert werden. Die Hauptaufgabe besteht darin, sowohl die moglichenUnterschiede zwischen den verschiedenen Eingangsparametern als auch die Anzahl von Parameternund deren Nutzen zur Informationsgewinnung zu untersuchen. Demzufolge wurden die Eingangspa-rameter hinsichtlich ihrer Klassifikationsgenauigkeit auf vorhandene SAR Daten verglichen. In dervorgeschlagenen Untersuchung stellen die Genauigkeitsanalysen ein objektives Kriterium fur den so-genannten “Nutzen” der Eingangsparameter dar.

Im zweiten Teil wurden die zuvor genannten polarimetrischen Eingangsparameter fur die interfer-ometrische Anwendung eingesetzt. Im Vordergrund stand die Bestimmung der Volumenstreuung undderen Einfluss auf eines der Elemente der interferometrischen Koharenz. Auch hier fand ein Vergle-ich der ausgesuchten Parameter in Bezug auf ihre Fahigkeit, den Effekt der Volumenstreuung in derinterferonmetrischen Koharenz zu reduzieren, statt.

Diese Arbeit mochte eine allgemeine Erfassung von polarimetrischen Eingangsparametern geben,wobei ein wichtiger Punkt, vom Autor hoffentlich erreicht, die Vollstandigkeit ist. Die doppelte Sichtder vorgestellten Untersuchungen, angelehnt an die Polarimetrie und Interferometrie, tragt zu einemumfassenden Verstandnis der Parameter in dieser Arbeit bei.

Stichworte: Abbildendes Radar, Synthetisches Apertur Radar (SAR), SAR-Polarimetrie, SAR-Interferometrie,Polarimetrische SAR-Interferometrie, Klassifikation.

“Tan cerca que tu mano sobre mi pecho es mıa”Pablo Neruda

“...It’s never over,My kingdom for a kiss upon her shoulderIt’s never over,All my riches for her smiles when I slept so soft against herIt’s never over,All my blood for the sweetness of her laughter...”

Jeff Buckley

Acknowledgements

This is a report of the work done during the period from 1999 to 2002 at the Microwavesand Radar Institute of the German Aerospace Centre (DLR) in Oberpfaffenhofen, Ger-many.

The results here collected would have not been possible without the help of a greatnumber of people. Foremost, I am indebted to my advisors Prof. Madhu Chandra andProf. Gerd Wanielik from the Technical University of Chemnitz. The former was mygroup leader at DLR during my first three years there in the frame of the EU fundedTraining and Mobility of Researchers (TMR) project that he directed. He accepted theapplication for a pre-doc position from a guy from a never-heard-of-before town inSouthern Italy and supported his work with a lot of “a priori” trust in his capabilities. Iam very grateful for that. The latter agreed to take me as PhD fellow and gave me thechance to make further profit of my work in Oberpfaffenhofen.

Sincere thanks to the people that permitted and sustained my stay at DLR: Dr. Wolf-gang Keydel and Prof. Alberto Moreira that followed each other as directors of theInstitute and always guaranteed their support; Dr. David Hounam and Rudi Schmid,who practically took charge of my position within the EU project and Marian Wernerwho offered me the opportunity of a new interesting job and to complete this thesis.

I wish to thank those people who helped me in revising the manuscript and whotaught me, or at least tried to teach me, something (my ability in forgetting things isastonishing! ): Dr. Ernst Luneburg for carefully going through the whole work andfundamentally influencing its writing, I regret I could not follow all his suggestionsfor further research, we should have started our collaboration earlier; Dr. Jose LuisAlvarez-Perez, “Padre Alvarez”, for his kindness and attention, I almost do not remem-ber a single “ I don’t know” or “ I can’t help you” from him; Dr. Kostas Papathanas-siou, for his explanations on polarimetric SAR, interferometry and penguin hunting, healso gave me several practical suggestions regarding this and other works; Dr. JurgenHolzner for his comments on the chapter on interferometry; Dr. Eleni Paliouras fora final English grammar check. Thanks also to Dr. Ernst Krogager from the DanishDefence Research Establishment (DDRE) in Copenhagen and to Dr. William Cameronfrom Boeing Defence and Space Group in Seattle for the discussions on target decom-position theory.

A special thank goes to Annette Wachter, Luis Ruby, Andrea Pelizzari and Dr. KatjaLamprecht for their German first-aid help, to Marco Quartulli and Herbert Daschielfor technical troubleshooting and for their company at coffee breaks and to Luca Pipiafrom the University of Cagliari (Italy, not Canada!), who tolerated me as graduationthesis supervisor, for the long talks on science and other (less “noble”) subjects.

ix

x Acknowledgements

Thanks to Dr. Irena Hajnsek for providing me with some figures, to Ralf Horn forthe picture of the DLR aeroplane and to Dr. Valeria Radicci from the University of Barifor helping me in formatting them. I am also grateful to Birgit Wilhelm for savingdata, burning CDs, printing slides, etc. on many occasions and to Laura Carrea (Tech-nical University of Chemnitz) for solving all the bureaucratic questions concerning thepresentation of this work and the academic title.

A mention is worth for Prof. Francesco Posa from the University of Bari and forAlma Blonda and Giuseppe Satalino from the Institute of Intelligent Systems for Au-tomation (ISSIA) of the National Research Council (CNR) in Bari, through them I wasintroduced to remote sensing and image classification: everything started with them.

I do not want to forget those people that “simply” offered me their friendship, ormore, during this adventure abroad: Sonia, Silvia, Antonio, Millah, Zsofi, Dani, San-dro, Ana Paula, Kais, Sara, Andrea, Christine, Paolo, Jose, Rafael, Josef, Adele, Licio,Axel, Sascha, Michael, Heike, Ewan, Carlos, Emiliano (I really hope nobody is miss-ing).

Alla mia famiglia va la mia gratitudine per avermi sempre e comunque appoggiato.

Thanks, again, to everybody,

Wolf

Contents

1 Introduction 1

2 Foundations of radar polarimetry 72.1 Description of electromagnetic waves . . . . . . . . . . . . . . . . . . . . 7

2.2 Scattering of electromagnetic waves . . . . . . . . . . . . . . . . . . . . . 16

2.3 Change-of-basis theory and characteristic polarizations . . . . . . . . . . 19

2.4 Scattering vectors and second order matrices . . . . . . . . . . . . . . . . 23

2.5 Target decomposition theorems . . . . . . . . . . . . . . . . . . . . . . . . 26

3 SAR polarimetry 313.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 SAR sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2 Early experimental results . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.3 Calibration techniques . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Interactions with the Earth surface . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Entropy/α analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Polarimetric SAR interferometry . . . . . . . . . . . . . . . . . . . . . . . 40

4 Analysis of polarimetric parameters: image classification 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Classification theory and accuracy assessment . . . . . . . . . . . . . . . 48

4.2.1 Classification theory basics . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Parallelepiped classification . . . . . . . . . . . . . . . . . . . . . . 49

4.2.3 Maximum likelihood classification . . . . . . . . . . . . . . . . . . 50

4.2.4 Minimum distance classification . . . . . . . . . . . . . . . . . . . 52

4.2.5 Classification accuracy assessment . . . . . . . . . . . . . . . . . . 52

4.3 Overview on polarimetric parameters . . . . . . . . . . . . . . . . . . . . 54

4.4 Experimental approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Data sets and test areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Backscattered wave amplitude . . . . . . . . . . . . . . . . . . . . . . . . 59

xi

xii Contents

4.7 Ratios of the scattering matrix elements . . . . . . . . . . . . . . . . . . . 61

4.8 Characteristic polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.9 Parameters of the coherent target decomposition theorems . . . . . . . . 66

4.10 Entropy/α parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.11 Comparisons and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Analysis of polarimetric parameters: interferometry 795.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Theoretical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1 Interferograms generation . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 Decorrelation sources . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.3 Interferometric coherence enhancement . . . . . . . . . . . . . . . 84

5.3 Interferometric coherence analysis . . . . . . . . . . . . . . . . . . . . . . 88

5.3.1 General correlation properties . . . . . . . . . . . . . . . . . . . . 88

5.3.2 Volume decorrelation . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 Interferometric phase analysis . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5 Final considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Conclusions 107

A Relationships among polarization geometrical parameters 109

B Target decomposition theorems 113B.1 Krogager decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B.2 Cameron decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

C Classification results 119C.1 Backscattered wave amplitude . . . . . . . . . . . . . . . . . . . . . . . . 120

C.2 Characteristic polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C.3 Parameters of the coherent target decomposition theorems . . . . . . . . 132

C.4 Entropy/α parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Bibliography 163

List of Figures

1.1 Overview of spaceborne SAR missions. . . . . . . . . . . . . . . . . . . . 3

2.1 General bistatic scattering geometry and local coordinate systems. . . . . 10

2.2 Detailed view of the transmitting local coordinate system, with projec-tion in the h1v1-plane of the track described by the tip of the ~ET vector(with the hypothesis that the relative phase δT of the electric field com-ponents is different from 0). . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Polarization ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 The Poincare sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 The Deschamps sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 SAR imaging geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Scattering behaviour relative to: (a) smooth, (b) rough, (c) very roughsurfaces (courtesy of I. Hajnsek [Haj01]). . . . . . . . . . . . . . . . . . . . 36

3.3 Diagram for determining the phase difference between two parallel wavesscattered from different points on a rough surface (courtesy of I. Hajnsek[Haj01]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Scheme of the α angle interpretation. . . . . . . . . . . . . . . . . . . . . . 39

3.5 H/α plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Interferometric SAR imaging geometry. . . . . . . . . . . . . . . . . . . . 41

3.7 Interferometric SAR imaging creation (courtesy of M. Quartulli [Qua98]). 42

4.1 The DLR Dornier DO 228-212 aircraft and the displacement of the dif-ferent SAR antennae mounted on board (courtesy of R. Horn). . . . . . . 57

4.2 Optical picture of the Oberpfaffenhofen test site (the yellow rectangledefines the area corresponding to the SAR data). . . . . . . . . . . . . . . 58

4.3 Accuracy estimation of the classification tests based on the amplitudesof the three [S] matrix elements: (a) overall accuracy; (b) Kappa (October‘99). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Classification maps based on the amplitudes of the three [S] matrix el-ements (15×15-pixel averaging window): (a) maximum likelihood; (b)minimum distance; (c) parallelepiped (October ‘99). . . . . . . . . . . . . 60

4.5 ZDR image of the Oberpfaffenhofen test site (October ‘99). . . . . . . . . 62

xiii

xiv List of Figures

4.6 Mean values and standard deviations of the polarization ratios in theselected regions of interest: (a) Munich; (b) Oberpfaffenhofen (October‘99). The ratios have been calculated, respectively, for the values relativeto a 5×5-pixel averaging window and on a single-pixel basis. . . . . . . 63

4.7 Accuracy estimation of the classification tests based on the co- andcross-polar nulls terms: (a) overall accuracy; (b) Kappa (October ‘99). . . 64

4.8 Classification maps based on the co- and cross-polar nulls terms (15×15-pixel averaging window): (a) maximum likelihood; (b) minimum dis-tance; (c) parallelepiped (October ‘99). . . . . . . . . . . . . . . . . . . . . 65

4.9 Accuracy estimation of the classification tests based on the three coef-ficients of the Krogager decomposition: (a) overall accuracy; (b) Kappa(October ‘99). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.10 Accuracy estimation of the classification tests based on the three co-efficients of the Pauli decomposition: (a) overall accuracy; (b) Kappa(October ‘99). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.11 Accuracy estimation of the classification tests based on the norm of thetwo scattering vectors representing the Cameron decomposition terms:(a) overall accuracy; (b) Kappa (October ‘99). . . . . . . . . . . . . . . . . 67

4.12 Classification maps based on the Krogager decomposition terms (15×15-pixel averaging window): (a) maximum likelihood; (b) minimum dis-tance; (c) parallelepiped (October ‘99). . . . . . . . . . . . . . . . . . . . . 68

4.13 Classification maps based on the Pauli decomposition terms (15×15-pixel averaging window): (a) maximum likelihood; (b) minimum dis-tance; (c) parallelepiped (October ‘99). . . . . . . . . . . . . . . . . . . . . 69

4.14 Classification maps based on the Cameron decomposition terms (15×15-pixel averaging window): (a) maximum likelihood; (b) minimum dis-tance; (c) parallelepiped (October ‘99). . . . . . . . . . . . . . . . . . . . . 70

4.15 H/α plane and its division in sub-regions according to the scatteringtypes (courtesy of I. Hajnsek). . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.16 Accuracy estimation of the classification tests based on the H and αparameters: (a) overall accuracy; (b) Kappa (October ‘99). . . . . . . . . . 73

4.17 Accuracy estimation of the classification tests based on the H , α and Aparameters: (a) overall accuracy; (b) Kappa (October ‘99). . . . . . . . . . 73

4.18 Classification maps based on the H and α parameters (10 × 10-pixelaveraging window): (a) maximum likelihood; (b) minimum distance;(c) parallelepiped (October ‘99). . . . . . . . . . . . . . . . . . . . . . . . . 74

4.19 Classification maps based on the H , α and A parameters (10×10-pixelaveraging window): (a) maximum likelihood; (b) minimum distance; (c)parallelepiped (October ‘99). . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 Range history R(R′, x−x′) of a point scatterer belonging to an extendedtarget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Range coordinate system for a fixed point in azimuth. . . . . . . . . . . . 81

List of Figures xv

5.3 Front view of the interferometric data-take geometry. . . . . . . . . . . . 82

5.4 Schematic representation of the random volume over ground scatteringmodel proposed in [PC01]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.5 Interferometric coherence derived from: (a) the Shh elements of thesphere term; (b) the Shh elements of the diplane term; (c) the Shh ele-ments of the helix term; (d) the unitary polarization vectors represent-ing the sphere term; (e) the unitary polarization vectors representing thediplane term; (f) the unitary polarization vectors representing the helixterm (May ‘98; baseline: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6 Interferometric coherence derived from the Shh elements of the diplaneterm (October ‘99; baseline: 12 m). . . . . . . . . . . . . . . . . . . . . . . 90

5.7 Interferometric coherence derived from: (a) the 1st Pauli term; (b) the2nd Pauli term; (c) the 3rd term; (d) the scattering vectors of Cameronmost dominant symmetric term; (e) the scattering vectors of Cameronleast dominant symmetric term (May ‘98; baseline: 15 m). . . . . . . . . . 92

5.8 Interferometric coherence derived from: (a) the 1st optimal value; (b)the 2nd optimal value; (c) the 3rd optimal value; (d) the original Shh

elements; (e) the original Shv elements; (f) the original Svv elements (May‘98; baseline: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.9 Histograms of the interferometric coherence of the whole area derivedfrom: (a) the Shh of the SDH decomposition terms; (b) the scattering vec-tors of the SDH decomposition terms; (c) the Pauli decomposition coef-ficients; (d) the scattering vectors of the Cameron decomposition terms;(e) the optimal polarizations; (f) the original polarimetric data (May ‘98;baseline: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.10 Histograms of the interferometric coherence for urban areas derivedfrom: (a) the Shh of the SDH decomposition terms; (b) the scattering vec-tors of the SDH decomposition terms; (c) the Pauli decomposition coef-ficients; (d) the scattering vectors of the Cameron decomposition terms;(e) the optimal polarizations; (f) the original polarimetric data (May ‘98;baseline: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.11 Histograms of the interferometric coherence for forested areas derivedfrom: (a) the Shh of the SDH decomposition terms; (b) the scattering vec-tors of the SDH decomposition terms; (c) the Pauli decomposition coef-ficients; (d) the scattering vectors of the Cameron decomposition terms;(e) the optimal polarizations; (f) the original polarimetric data (May ‘98;baseline: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.12 Interferometric phases of: (a) the Shh elements of the sphere term; (b)the Shh elements of the diplane term. (c) Difference of the two interfero-metric phases (May ‘98, baseline: 15 m). . . . . . . . . . . . . . . . . . . . 100

5.13 Interferometric phases of: (a) the unitary polarization vectors represent-ing the sphere term; (b) the unitary polarization vectors representing thehelix term. (c) Difference of the two interferometric phases (May ‘98,baseline: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

xvi List of Figures

5.14 Interferometric phases of: (a) the 1st Pauli term; (b) the 2nd Pauli term.(c) Difference of the two interferometric phases (May ‘98, baseline: 15 m). 102

5.15 Interferometric phases of: (a) the scattering vectors of Cameron mostdominant symmetric term; (b) the scattering vectors of Cameron leastdominant symmetric term. (c) Difference of the two interferometric phases(May ‘98, baseline: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.16 Interferometric phases of: (a) the 1st optimal value; (b) the 2nd optimalvalue. (c) Difference of the two interferometric phases (May ‘98, base-line: 15 m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A.1 Polarization ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

List of Tables

4.1 Confusion matrix relative to the Min. Dist. classification test performedon the amplitudes of the three [S] matrix elements, with a 15 × 15-pixelaveraging window (values in percentage). . . . . . . . . . . . . . . . . . . 53

4.2 Main E-SAR system parameters. . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Accuracy estimates of all the classification tests relative to the 15×15-pixel averaging window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

C.1 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the the three [S] matrix elements (single-pixel basis). . . . 120

C.2 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed the three [S] matrix elements (single-pixel basis). 120

C.3 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the the three [S] matrix elements (single-pixel basis). . 121

C.4 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed the three [S] matrix elements (single-pixel basis). 121

C.5 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the the three [S] matrix elements (single-pixel basis). . . . 122

C.6 Commission and omission error estimates relative to the Parall. classifi-cation test performed the three [S] matrix elements (single-pixel basis). . 122

C.7 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the the three [S] matrix elements (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

C.8 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed the three [S] matrix elements (15×15-pixel av-eraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

C.9 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classification testperformed on the the three [S] matrix elements (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C.10 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed the three [S] matrix elements (15×15-pixel av-eraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C.11 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the the three [S] matrix elements (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xvii

xviii List of Tables

C.12 Commission and omission error estimates relative to the Parall. classifi-cation test performed the three [S] matrix elements (15×15-pixel averag-ing window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

C.13 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the [S] matrix co- and cross-polar nulls (single-pixel basis). 126

C.14 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the [S] matrix co- and cross-polar nulls (single-pixel basis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C.15 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the [S] matrix co- and cross-polar nulls (single-pixelbasis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

C.16 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the [S] matrix co- and cross-polar nulls (single-pixel basis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

C.17 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the [S] matrix co- and cross-polar nulls (single-pixel basis). 128

C.18 Commission and omission error estimates relative to the Parall. classifi-cation test performed on the [S] matrix co- and cross-polar nulls (single-pixel basis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

C.19 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classificationtest performed on the [S] matrix co- and cross-polar nulls (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

C.20 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the [S] matrix co- and cross-polar nulls (15×15-pixel averaging window). . . . . . . . . . . . . . . . . . . . . . . . . . . 129

C.21 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the [S] matrix co- and cross-polar nulls (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

C.22 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the [S] matrix co- and cross-polar nulls (15×15-pixel averaging window). . . . . . . . . . . . . . . . . . . . . . . . . . . 130

C.23 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the [S] matrix co- and cross-polar nulls (15×15-pixel aver-aging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

C.24 Commission and omission error estimates relative to the Parall. classifi-cation test performed on the [S] matrix co- and cross-polar nulls (15×15-pixel averaging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

C.25 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the SDH decomposition terms (single-pixel basis). . . . . . 132

C.26 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the SDH decomposition terms (single-pixelbasis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

List of Tables xix

C.27 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the SDH decomposition terms (single-pixel basis). . . 133

C.28 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the SDH decomposition terms (single-pixelbasis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

C.29 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the SDH decomposition terms (single-pixel basis). . . . . . 134

C.30 Commission and omission error estimates relative to the Parall. classi-fication test performed on the SDH decomposition terms (single-pixelbasis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

C.31 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classificationtest performed on the SDH decomposition terms (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

C.32 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the SDH decomposition terms (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

C.33 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the SDH decomposition terms (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

C.34 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the SDH decomposition terms (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

C.35 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the SDH decomposition terms (15× 15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

C.36 Commission and omission error estimates relative to the Parall. classi-fication test performed on the SDH decomposition terms (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

C.37 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the Pauli decomposition terms (single-pixel basis). . . . . 138

C.38 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the Pauli decomposition terms (single-pixelbasis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C.39 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the Pauli decomposition terms (single-pixel basis). . . 139

C.40 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the Pauli decomposition terms (single-pixelbasis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.41 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the Pauli decomposition terms (single-pixel basis). . . . . 140

C.42 Commission and omission error estimates relative to the Parall. classi-fication test performed on the Pauli decomposition terms (single-pixelbasis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

xx List of Tables

C.43 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the Pauli decomposition terms (15× 15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.44 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the Pauli decomposition terms (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.45 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the Pauli decomposition terms (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

C.46 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the Pauli decomposition terms (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

C.47 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the Pauli decomposition terms (15× 15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

C.48 Commission and omission error estimates relative to the Parall. classi-fication test performed on the Pauli decomposition terms (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

C.49 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the Cameron decomposition terms (single-pixel basis). . . 144

C.50 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the Cameron decomposition terms (single-pixel basis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

C.51 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the Cameron decomposition terms (single-pixel basis). 145

C.52 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the Cameron decomposition terms (single-pixel basis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

C.53 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the Cameron decomposition terms (single-pixel basis). . . 146

C.54 Commission and omission error estimates relative to the Parall. classi-fication test performed on the Cameron decomposition terms (single-pixel basis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

C.55 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the Cameron decomposition terms (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

C.56 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the Cameron decomposition terms (15×15-pixel averaging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

C.57 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classification testperformed on the Cameron decomposition terms (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

List of Tables xxi

C.58 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the Cameron decomposition terms (15×15-pixel averaging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

C.59 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the Cameron decomposition terms (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

C.60 Commission and omission error estimates relative to the Parall. classifi-cation test performed on the Cameron decomposition terms (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

C.61 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the H and α parameters (3×3-pixel averaging window). . 150

C.62 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on theH and α parameters (3×3-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C.63 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the H and α parameters (3×3-pixel averaging window).151

C.64 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on theH and α parameters (3×3-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

C.65 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the H and α parameters (3×3-pixel averaging window). . 152

C.66 Commission and omission error estimates relative to the Parall. classifi-cation test performed on the H and α parameters (3×3-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

C.67 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the H and α parameters (15×15-pixel averaging window). 153

C.68 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the H and α parameters (15×15-pixel aver-aging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

C.69 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classification testperformed on the H and α parameters (15×15-pixel averaging window). 154

C.70 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the H and α parameters (15×15-pixel aver-aging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

C.71 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the H and α parameters (15×15-pixel averaging window). 155

C.72 Commission and omission error estimates relative to the Parall. classifi-cation test performed on theH and α parameters (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

C.73 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on the H , α and A parameters (3×3-pixel averaging window). 156

xxii List of Tables

C.74 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the H , α and A parameters (3×3-pixel aver-aging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

C.75 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classification testperformed on the H , α and A parameters (3×3-pixel averaging window). 157

C.76 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the H , α and A parameters (3×3-pixel aver-aging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

C.77 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on the H , α and A parameters (3×3-pixel averaging window). 158

C.78 Commission and omission error estimates relative to the Parall. classifi-cation test performed on theH , α andA parameters (3×3-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

C.79 Pr. Acc. and Us. Acc. estimates relative to the Max. Lik. classification testperformed on theH , α andA parameters (15×15-pixel averaging window).159

C.80 Commission and omission error estimates relative to the Max. Lik. clas-sification test performed on the H , α and A parameters (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

C.81 Pr. Acc. and Us. Acc. estimates relative to the Min. Dist. classificationtest performed on the H , α and A parameters (15×15-pixel averagingwindow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

C.82 Commission and omission error estimates relative to the Min. Dist. clas-sification test performed on the H , α and A parameters (15×15-pixelaveraging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

C.83 Pr. Acc. and Us. Acc. estimates relative to the Parall. classification testperformed on theH , α andA parameters (15×15-pixel averaging window).161

C.84 Commission and omission error estimates relative to the Parall. classifi-cation test performed on the H , α and A parameters (15×15-pixel aver-aging window). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

1 Introduction

Research in the field of radar remote sensing has been a promising area for severaldecades. In particular, imaging radar systems have become a powerful tool for study-ing the Earth and its environment. The reasons which make radars useful may besummarized as follows:

• radars are active systems: this means that they generate electromagnetic (EM)wave beams whose scattering by targets is the object of study and that they canoperate independently of daylight (unlike, for example, optical sensors).

• Frequency, polarization, power and direction of the transmitted beam can be cho-sen at will. Radars operate within the microwave (MW) region of the EM spec-trum that ranges from frequencies of about 3 MHz up to 300 GHz, with corre-sponding wavelengths λ from 100 m down to 1 mm. However, most of the civilsystems nowadays in use limit themselves to a main set of frequency bands: X(f ≈ 10 GHz or λ ≈ 3 cm), C (f ≈ 6 GHz or λ ≈ 5 cm), S (f ≈ 3 GHz or λ ≈ 10cm), L (f ≈ 2 GHz or λ ≈ 15 cm) and P (f ≈ 0.5 GHz or λ ≈ 60 cm).

• Radars can provide (for certain values of moisture and density of the ground)information from beneath the ground surface (sub-surface information). In thesame way, they can go through vegetation canopies and give information on theircharacteristics and on the terrain beneath. Indeed, the penetration depth of anEM wave is a function of the density and moisture content of the penetratedmedium as well as of the frequency and polarization of the wave itself.

• In most of the common radar frequency bands, these sensors are almost indepen-dent of weather conditions, as the attenuation of the atmosphere is negligible forwavelengths λ > 3 cm.

Synthetic aperture radar (SAR) systems can provide better results than conventionaldirect aperture radars. Whereas the resolution of an airborne or spaceborne radar de-pends on antenna dimensions and on the distance from the targets, SAR devices cansimulate antenna dimensions much larger than the real ones and can make the resolu-tion along the flight-path direction independent from the sensor-to-target distance.

Conventional imaging radars (including SAR) operate with a single fixed-polarization antenna for both transmission and reception of radar signals. In this way,for each resolution element in the image, a single scattering coefficient is measured fora specific combination of transmit and receive polarization states used for recordingthe radar echo; hence, a scalar processing is applied to the power backscattered from

1

2 Chapter 1 - Introduction

the observed targets. The use of polarization-sensitive devices is the logical develop-ment that follows from the consideration of the vector nature of the EM waves. Fullypolarimetric radars are indeed able to transmit and receive both the orthogonal com-ponents of an EM wave. This ensures that the complete scattering information carriedby radar echo signals is fully used to enhance targets detection and identification.

A brief digression on polarimetry history, covering the last 50 years, would now beuseful.

Although the discovery of polarimetric phenomena in light dates back to the sev-enteenth century, the earliest studies on polarimetric radars appeared at the end ofthe 40s. Worth mentioning are in particular those reported by Kennaugh [Ken54], Sin-clair [Sin50], Deschamps [Des51], Graves [Gra56] and Copeland [Cop60] which antic-ipated the classic PhD work by Huynen [Huy70] in 1970. Kennaugh’s contributionwas probably the most meaningful; unfortunately it was for several years classifiedand only at the end of the 70s made available for the scientific community. A detailedreview of all these contributions was later conducted by Boerner [BEACM81], [eae85],[eae92]. However, the potential of radar polarimetry remained underestimated untilthe end of the 80s, mainly because of technological limits restricting practical appli-cations. A fundamental turning point was represented by the NASA/JPL airborneAIRSAR [ZvZH87], [ZvZ91] that was the first imaging polarimetric SAR system everoperated and the forerunner to a series of others by different research institutions (seeFigure 1.1).

Thus, concrete applications of polarimetric techniques became possible only recently,when developments in technology (among others the availability of digital data record-ing systems and general purpose computers for data reduction) permitted the imple-mentation of fully polarimetric imaging radars. Even more important has been the so-lution of the problems related to the coherent analysis of the waves, i. e., the exact mea-surement of the signal phase. With regard to the role played by phase, backscatteredsignals depend completely on the nature of the targets and two extreme situations arepossible: point scatterers and Gaussian scatterers. In the first case, the resolution cellmay be treated as point-like when determining its position and no uncertainties inphase are present. Backscattering from Gaussian scatterers is, on the contrary, due to anumber of elementary random scatterers among which none provides a contributionclearly dominating the others. The evaluation of the phase is in this case the result ofan integration over the various contribution from the chosen resolution cell.

It is currently possible to distinguish different branches of what are generally calledpolarimetric studies. Regardless of the common basis they share, there are troublesin relating them partly due to the different conventions adopted in radar and opticalpolarimetry. For example, the very definition of polarization handedness can lead toconfusion and misunderstandings.1

1“. . . Circularly polarized waves have either a right-handed polarization or a left-handed polariza-tion, which is defined by convention. The TELSTAR satellite sent out circularly polarized microwaves.When it first passed over the Atlantic, the British station at Goonhilly and the French station at PleumeurBodou both tried to receive its signals. The French succeeded, because their definition of sense of po-larization agreed with the American definition. The British station was set up to receive the wrongpolarization because their definition of sense of polarization was contrary to our definition. . . ”

from J. R. Pierce, “Almost everything about waves”. Cambridge, Massachusetts USA; MIT Press, 1974,pages 130-131.

Chapter

1-Introduction

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4 Chapter 1 - Introduction

remote sensing systems. In this context, the EM waves may be described eitherusing two-dimensional Jones vectors or four-dimensional Stokes vectors and, ac-cordingly, their scattering interactions will be represented by means of scatteringor Stokes matrices. The theory explaining the relations among different represen-tations and the operations of change-of-basis also have a general relevance.

SAR polarimetry. As anticipated, the principles of synthetic aperture have been ap-plied also to polarimetric radars giving new momentum to the microwave remotesensing of the Earth. In some cases, the full scattering matrices or the Kennaughmatrices can be measured. Technological aspects as well as those connected toapplications of the data are developing rapidly. Much attention is paid, for ex-ample, to calibration and signal processing or to classification and pattern recog-nition methods.

Scatterer property modelling. This is the “ultimate” part of the research, which in-volves both theoretical and experimental aspects: modelling the scattering be-haviour of point and distributed targets and the interactions using few parame-ters only, and developing algorithms to invert the models for retrieving physicalquantities from the measurements.

Polarimetric interferometry. A recent development of polarimetry arises from its as-sociation with interferometric techniques. Their combined application yields no-table results: polarimetric analysis makes it possible to separate different scat-tering mechanisms within a given resolution cell, whereas interferometric tech-niques allow for a topographical characterization of the scattering contributions.These properties can then be used for improving digital elevation models (DEM)or for biomass estimation.

Finally, tomographic and holographic techniques are also applicable to SAR data[RM00], [Rei01] and can be combined with interferometric tools, as they share the com-mon goal of achieving elevation maps or 3-D images of the observed scenes.

The following two chapters will be devoted to a review, as complete as possible, ofthe theory concerning these and other issues. The aim is to define, for each branch, theestablished knowledge and the possible future developments of research.

The experimental part of this work will pursue another task. Given the differentpossibilities of expressing polarimetric data, we will consider them in a systematic way,trying to see if substantial differences exist among the various polarimetric observables in termsof the amount of information they can provide and of its usefulness. For this reason, in Chap-ter 4 we have compared a set of these quantities by means of measures of classificationaccuracy. In other words, we have tested different classification algorithms on ob-servables extracted by airborne polarimetric SAR data. The chosen algorithms are notspecifically suited for this kind of data but, considering our approach, this is only ofsecondary importance. Indeed, with our tests we want to have a measure (hence, some ob-jective quantities) of the “utility” of the studied observables. A further reason for this choicewas that the classification algorithms used are well-established and in general use inthe remote sensing scientific community. In this way we may also get some idea of thepotential for wider use of polarimetric SAR data.

Chapter 1 - Introduction 5

Following this systematic overview, we will consider in Chapter 5 some of the po-larimetric observables and examine them in combination with interferometry. The goalis to determine whether in this way a characterization of volume scattering, one of the termsaffecting the interferometric coherence, is possible. Again, a comparison of the chosen polari-metric quantities has been done and an attempt to estimate their usefulness is performed.

As indicated by its general scheme, this work is intended as a survey of polarimetricobservables and one of its merits is hopefully its completeness. Finally, to provide abetter understanding about the observables, these have been considered from differentpoints of view, namely in classification and SAR interferometry applications.

2 Foundations of radar polarimetry

2.1 Description of electromagnetic waves

Comprehensive introductions to polarimetry theory may be found in classics by Bornand Wolf [BW85], Kennaugh [Ken52], [Ken54], Huynen [Huy70], Azzam and Bashara[AB77] and Mott [Mot92]. Also a great number of journal publications have been dedi-cated during the years to general theoretical aspects; noteworthy examples are those byCloude [Clo83], [Clo86], van Zyl et al. [vZZE87] and by Kostinski and Boerner [KB86]whose title we borrow for this chapter. These works are all interesting descriptions ofthe state of the art in this field during the eighties and collect almost all the basic equa-tions of polarimetry. Other reviews of the main concepts of radar polarimetry havebeen later presented in [AB89], [vZZ90], [BYXY91] and [Hub94].

As a first step we will introduce EM waves and see how they can be representedand how different representations are related to each other.

All aspects of macroscopic electromagnetic phenomena may be described in termsof the set of the Maxwell equations that in MKSA units have the form [Str41], [Jac75],[Kon86]:

~∇× ~E(~r, t) = − ∂

∂t~B(~r, t), (2.1)

~∇× ~H(~r, t) = ~J(~r, t) +∂

∂t~D(~r, t), (2.2)

~∇ · ~B(~r, t) = 0 , (2.3)~∇ · ~D(~r, t) = % (~r, t), (2.4)

where:

• ~E is the electric field intensity vector in V olt/meter,

• ~B is the magnetic flux density vector in Tesla,

• ~H is the magnetic field intensity vector in Ampere/meter,

• ~D is the current displacement vector in Coulomb/meter2,

• ~J is the electric current density vector in Ampere/meter2 and

• % is the electric charge density in Coulomb/meter3.

7

8 Chapter 2 - Foundations of radar polarimetry

~E, ~B, ~H , ~D, ~J and % are all real-valued functions of time t and spatial location ~r, with~r being a position vector defined with respect to a specified coordinate system.

If the field vectors are linearly related, the medium is said to be linear and one has:

~B(~r, t) = µ(~r) ~H(~r, t) , (2.5)~D(~r, t) = ε(~r) ~E(~r, t) , (2.6)

indicating with µ(~r) (expressed in Farad/meter) and ε(~r) (in Henry/meter), respec-tively the dielectric tensor and the magnetic permeability tensor of the medium. In ahomogeneous medium µ, and ε are constant and in an isotropic medium they are scalarquantities.

Generation, propagation and interactions of EM waves with matter are governed bythese laws. Indeed, by means of simple combinations of the Maxwell equations andusing further relationships among the above defined quantities, one can prove that forthe vector fields in a homogeneous isotropic medium a wave motion equation holdssuch as:

∇2~Ψ(~r, t) − 1

v2

∂2

∂t2~Ψ(~r, t) = ~g(~r, t) , (2.7)

with ~Ψ being one of the fields, v the wave propagation velocity and ~g a function of thesources generating the wave.1

In a linear, source-free (i. e., for % (~r, t) = 0 and in absence of externally appliedelectric currents), homogeneous isotropic medium, the wave equation is homogeneousfor every component of ~E:

∇2 ~E(~r, t) − 1

v2

∂2

∂t2~E(~r, t) = 0 , (2.12)

with:v =

1√µε. (2.13)

1Let us show how an equation similar to (2.7) can be derived for the electric field ~E in a homogeneousisotropic medium. From Equations (2.1) and (2.5) it follows:

~∇× (~∇× ~E(~r, t)) = ~∇×(

− ∂

∂t~B(~r, t)

)

= ~∇×(

−µ∂

∂t~H(~r, t)

)

= −µ∂

∂t

(

~∇× ~H(~r, t))

. (2.8)

Applying the vector identity ~∇×(~∇×~a) = ~∇(~∇·~a)−∇2~a, and substituting ~∇× ~H(~r, t) from Equation (2.2)one has:

~∇(~∇ · ~E(~r, t)) −∇2 ~E(~r, t) = −µ∂

∂t~J(~r, t) − µ

∂2

∂t2~D(~r, t), (2.9)

that by means of (2.6) becomes:

∇2 ~E(~r, t) − ~∇(~∇ · ~E(~r, t)) − µε∂2

∂t2~E(~r, t) = µ

∂

∂t~J(~r, t). (2.10)

Using Equation (2.4) and assuming ~∇% (~r, t) = 0, Equation (2.10) may be written as:

∇2 ~E(~r, t) − µε∂2

∂t2~E(~r, t) = µ

∂

∂t~J(~r, t), (2.11)

known as the inhomogeneous scalar wave or Helmholtz equation.

2.1 - Description of electromagnetic waves 9

Equation (2.12) allows as a solution any function of the type:

~Ψ(~r, t) = ~Ψ+(ωt− ~k ·~r ) + ~Ψ−(ωt+ ~k ·~r ). (2.14)

~Ψ+ and ~Ψ− represent waves propagating in opposite directions and are twice differen-tiable functions of:

φ± = ωt∓ ~k ·~r, (2.15)

where ω and ~k are respectively the angular frequency and the propagation vector ofthe wave defined concordantly to the position vector; hence, a real-valued solution ofthe wave equation of the electric field may be written as:

~E(~r, t) = ~Ereal cos(ωt− ~k ·~r ) (2.16)

(here only one of the two terms of the sum in (2.14) has been considered).

For practical reasons, it is however customary to adopt a complex representation(this is always possible due to the linearity of the wave equation):

~E(~r, t) = ~E exp j(ωt− ~k ·~r ), (2.17)

taking both ~E(~r, t) and ~E as complex, and to assign physical meaning only to its realpart, Re[ ~E(~r, t)]. This is the convention that we too will use in the following.

For a given real vector ~k, one can determine a phase front of ~E(~r, t) by setting ~k ·~r =

constant and note that it coincides with a plane orthogonal to ~k. Indeed, with thiscondition the amplitude of the electric field is the same for all points on the plane andvaries in time, remaining constant on this surface. Such waves are known as planewaves and Equation (2.17) represents their general form.

Let us consider now a simple physical system such as the one in Figure 2.1,composed of a transmitting antenna, a target and a receiving antenna and let us de-fine a global Cartesian coordinate system, with basis vectors x, y and z, with its originwithin the target. The plane containing the directions of propagation of the transmittedand scattered waves is the scattering plane; referring to it, the complex transverse com-ponents of the electric field illuminating the scatterer are expressed in terms of a localright-handed coordinate system (h1, v1, n1), coincident with the transmitting antenna,defined so that the n1-axis is directed towards the target. In a similar way a local coordinatesystem (h2, v2, n2) is defined with its origin in the receiving system.

In the far field and for small (compared with distances r) targets, waves can betreated as planar; thus, the electric fields of three monochromatic (i. e., completelypolarized) waves may be expressed as follows:

~ET = (E2Th1

+ E2Tv1

)1/2[cosαT h1 + sinαT ejδT v1] · exp j(ωt− ~k ·~r1 + φT ), (2.18)

~ER = (E2Rh2

+ E2Rv2

)1/2[cosαR h2 + sinαR ejδR v2] · exp j(ωt+ ~k ·~r2 + φR), (2.19)

~A = (A2h2

+ A2v2

)1/2[cosαA h2 + sinαA ejδA v2] · exp j(ωt− ~k · ~r2 + φA), (2.20)

where, in all the equations, α = arctan(Ev/Eh) (Eh and Ev being the absolute valuesof the complex components of the electric field), δ and φ are the relative and absolute


x

y

z

n

h

scatterer

v

n h

v

1

1

1

22

2

transmittingantenna

receivingantenna

� � �&�� '��!��% � �� !��$� # �(#$!�� (� '�� . �� !�'� �� #$!��#$��&�� &� '�!�� .*� �

phases respectively, subscripts T and R stand for transmitter and receiver, subscripts1 and 2 specify the coordinate system and ~A, the antenna height, is the wave that thereceiving antenna would radiate in the +n2 direction if it acted as a transmitter (for abetter understanding of the terms involved in the equations above, see also Figure 2.2and later on in this paragraph).

The three waves propagate in the +n1, −n2 and +n2 directions respectively. In thecase of backscattering, ~ET ∝ ~A only if the same antenna is used for transmitting andreceiving. When this condition is held to be true, one can then write [Ken54]:

~ET =Z0It2λr

~A, (2.21)

where Z0 is the impedence of the medium, It the terminal antenna current, λ the wave-length and r = r1 = r2.

For every monochromatic EM wave, the tip of the vector representing the electricfield describes an ellipse on each generic fixed plane normal to the direction of propa-gation (i. e., when one considers the evolution in time). According to the IEEE standarddefinitions [IEE83], the polarization of a wave receding from an observer is denotedright-handed if the electric field vector appears to be rotating clockwise in this planeand left-handed if it appears to be rotating counterclockwise on the chosen perpendicu-lar plane. Therefore, assuming α = π/4 and δ = π/2, we may define ~A as a left-handedcircularly polarized wave (because of its +n2 propagation) and ~ER as a right-handedone (as it propagates along the −n2 direction).

Before going further with the description of the scattering process, it is useful tointroduce some alternative representations of the EM waves. Let ~E be a generic waveof the form:

~E = (E2h+ E2

v)1/2[cosα h+ sinα ejδ v] · exp j(ωt− ~k · ~r + φ). (2.22)


T

n 1

h 1

v 1

E

αT

� � �� !�� !�'"� .�� $� '�� # !�� #$�� &� '�!�� .��/� �� # � � �&'�� '��

h1v1 �� !�'�� "� �� $�!�#� �� (#��0� %"� /% � ��"� � � � �� "� ~ET

�� # � �� /� �� "�� "�� !�� !� � �� !��(�

δT�� # �$�(� #�� #$�&.��'��' � � � � �� ' � �$� �&.��

This equation, which is similar to the ones above, can also be rewritten as:

~E = (E2h+ E2

v)1/2[cosα ejδhh+ sinα ejδv v] · exp j(ωt− ~k · ~r ), (2.23)

when each component is expressed with its own absolute phase, i. e. considering:

δh = φ (2.24)

andδ = δv − δh = δv − φ. (2.25)

One example of the different ways to represent ~E is the Jones vector representation,which is related to the particular choice we made of the wave local coordinate system.This is obtained by writing the wave as a complex two-dimensional column vectorlike:

E = (E2h+ E2

v)1/2 ejφ

[

cosαsinα ejδ

]

= (E2h+ E2

v)1/2

[

cosα ejδh

sinα ejδv

]

. (2.26)

In fact, an isomorphism exists relating generic two-dimensional vectors to two-dimensional column vectors (for which we adopted respectively the notations: ~V andV). In many cases, the Jones vectors are defined as normalized vectors, hence dividing(2.26) by (E2

h+ E2v)

1/2.

The term in square brackets, the polarization state (PS) of the wave, completely deter-mines its polarization ellipse at a fixed point in space, but does not contain the direc-tion of propagation, which must be recovered from the exponent appearing in (2.22).


a

h

b

v

χ

ψ

� � �&�� !��(� � !� � �&' �� "� ��

Hence, the only polarimetric information which cannot be reconstructed from the Jonesvectors is the handedness, since handedness involves the definition of the direction ofpropagation.

To avoid the lack of consistency caused by this insufficiency, the Jones vectors canbe complemented by the subscripts “+” and “−” in order to distinguish between twodirections of propagation: E+ should indicate waves propagating in the +~k directionand E− in the −~k direction:

~E+ = E+ exp j(ωt− ~k · ~r ) (2.27)~E− = E− exp j(ωt+ ~k · ~r ). (2.28)

The vectors E± are known as directional Jones vectors [Gra56]. E+ and E− representthe same state of polarization referring to opposite propagation directions if they arerelated by the complex conjugation operation, i. e.:

E± = E∗∓ . (2.29)

It should be stressed that this relation is valid only for linear polarization bases.

The equation of the polarization ellipse can be derived from the general expression(2.22). This new representation of the generic wave ~E, a geometrical one (see Fig-ure 2.3), is characterized by two parameters expressing the ellipticity, i. e., the ratio ofthe minor semi-axis b to the major semi-axis a [BEACM81], [BW85]:

tanχ =b

a(2.30)


and inclination angle ψ of the major axis. These parameters are related to those of theJones vector by means of:2

tan 2ψ = tan 2α cos δ, (2.31)sin 2χ = sin 2α sin δ (2.32)

and using them, the PS of ~E (the first column vector that appears in (2.26)) can beexpressed as:

p =

[

cosψ − sinψsinψ cosψ

] [

cosχj sinχ

]

. (2.33)

Values of χ between −π/4 and +π/4, and values of ψ between 0 and +π (or equiv-alently between −π/2 and +π/2) are sufficient to describe all possible polarizationstates. According to the convention adopted above, a wave propagating in the +~kdirection is right-handed if −π/4 ≤ χ < 0 and left-handed if 0 < χ ≤ π/4.

It is now easier to understand the meaning of Equation (2.29): the conjugation ofthe Jones vectors implies a change in the sign of the phase difference δ = δv − δh and,via (2.32), in the sign of the ellipticity angle χ. Hence, the handedness of the PS alsochanges accordingly.

Another representation of the waves closely related to the one we have just intro-duced, is the Stokes vector3 representation. Such vectors are defined in the followingway:

g =

g0

g1

g2

g3

=

IQUV

def=

|Eh|2 + |Ev|2|Eh|2 − |Ev|22Re{E∗

hEv}2 Im{E∗

hEv}

, (2.34)

where Eh and Ev are the complex components of ~E.

Each component4 of g describes a characteristic of the EM wave, in detail:

• I represents the total wave intensity,

• Q is the difference between horizontal and vertical intensities,

• U is the difference between ±π2

linear polarizations (the tendency of the wave tobe ±π

2linear polarized),

• V is the difference of intensities between right and left polarizations (the ten-dency to be left- or right-handed)

and it can be shown that they are not independent and furthermore that, for a com-pletely polarized wave (which is the case we are hitherto considering), it results:

g0 = (g21 + g2

2 + g23)

1/2. (2.35)2Mathematical details on the derivation of Equations (2.31) and (2.32) are reported in Appendix A.3A word of caution in the use of the term “vector” is required when referring to the Stokes vectors.

Since the common rules of addition of vectors and of product by a scalar cannot be defined, the Stokesvectors are not really vectors.

4The definition of g chosen here has been widely used in optics [BW85] and also seems to havebecome common in radar polarimetry in recent years. The definition preferred by Huynen [Huy70] andothers [Giu86] is instead obtained by substituting as follows: g1 → g2, g2 → g3 and g3 → g1.


polarizations

g

g

g

2χ

3

2

1

2ψ

right-handedpolarizations

left-handed

� � �&�� "� � �&� ' #$!��

The relationship with the previous geometrical representation may be brought to-gether by using (2.31) and (2.32), so that (2.34) can be rewritten as:

g =

E2h + E2

v

E2h − E2

v

2EhEv cos δ2EhEv sin δ

= E2

1cos 2ψ cos 2χsin 2ψ cos 2χ

sin 2χ

. (2.36)

In (2.36), E2 = E2h + E2

v is the total intensity of the wave.

The Stokes vector can be represented graphically by reference to the Poincare sphere.

All the possible polarizations of a wave with total intensity E2 = g0 are describedin a three-dimensional Cartesian system (see Figure 2.4). Each PS is represented bya point on a sphere of radius g0 whose Cartesian coordinates are (g1, g2, g3). On thissphere the angles 2ψ and 2χ represent the longitude and the latitude of the point defin-ing the PS. The equator of the Poincare sphere thus represents linear polarizations, thepoles represent circular polarizations and all left-handed (right-handed) elliptical po-larizations are mapped onto the northern (southern) hemisphere. The extremes of eachdiameter correspond to a pair of orthogonal polarizations [IEE83].

As a final example of the different ways to describe an EM wave, we introduce theconcept of complex polarization ratio, defined as the ratio of the orthogonal complexelectric field components [AB77]:

ρdef= Ev/Eh = Ev e

jδv/Eh ejδh = tanα ejδ = |ρ| ejδ, (2.37)


2ψ

g

g

g

3

2

1

2χ

δ

2α

� � �&�� 0# ��!�.��"� � ��

where 0 ≤ α < π/2 and 0 ≤ δ < 2π. The two angles α and δ are now defined as De-schamps parameters [Des51] and determine the polarization state of a wave in terms ofthe amplitude ratio and phase difference. Since these parameters can also be geometri-cally represented by means of a sphere, the Deschamps sphere, it is easy to visualize therelationship between them and ψ and χ (see Figure 2.5). Using ρ , the two-dimensionalcolumn vector becomes:

E = ejδh(E2

h+ E2v)

1/2

√1 + ρρ∗

[

1ρ

]

. (2.38)

The previously mentioned “orthogonality” of complex vectors (referred to PSs orgeneric vectors) may be defined in terms of an inner product of the form:

〈a|b〉 ≡ a∗xbx + a∗yby , (2.39)

where a and b are two generic column vectors, such that they are orthogonal if:

〈a|b〉 = 0 (2.40)

Now, according to the complex polarization ratio representation, a new definitionof orthogonality of two PSs, denoted by m and n, can be introduced:

ρmρ∗n = −1 ⇐⇒ ρm = − 1

ρ∗n, (2.41)

by means of some very general mathematical considerations (Feynman related it to theanalogous condition for the angular coefficients of two real straight lines [Fey96]).


2.2 Scattering of electromagnetic waves

Taking into account all the definitions above, we can now continue with the descriptionof the interactions among the elements of the system in Figure 2.1.

Under the far field assumption, it has been shown that a general EM scattering pro-cess can be described in terms of a matrix equation of the form [Mot92]:

ER = limr/λ→∞

λ

(4π)1/2r[S]ET , (2.42)

where the distance between the receiving antenna and the target is denoted by r, λ isthe operating wavelength and [S] is a 2 × 2 complex matrix, i. e.:

[S] =

[

Sh2h1Sh2v1

Sv2h1Sv2v1

]

. (2.43)

[S] is the scattering matrix and completely defines the interaction; it introduces a map-ping connecting the vectors representing the incident and scattered waves. Its elementsare, in general, complicated and sensitive functions of frequency, target orientation andshape, relative orientation of the polarization planes in the bistatic case, etc. The diag-onal elements Sh2h1

, Sv2v1and the off-diagonal ones Sh2v1

, Sv2h1are called respectively

co- and cross-polarized components. Note that the fractional term causes [S] to be di-mensionless, but (2.42) can be expressed in different ways and [S] can also have thedimension of a length. However, this proportional term will be omitted from here onand (2.42) written in the simplified form:

ER = [S]ET . (2.44)

In the most general case, there are seven independent parameters in the bistaticscattering matrix: four amplitudes and three phases. Indeed, an overall absolute phasemay be neglected, since the power received from the scatterer is independent of thisphase. In the backscatter (or monostatic) case, one has that h1 ≡ h2 and v1 ≡ v2 andreciprocity dictates that Sh2v1

= Sv2h1and there are only five independent parameters

in the scattering matrix.

Another fundamental equation of radar polarimetry comes from a different elementof the system: the receiving antenna network. It relates the voltage measured at areceiving antenna to the polarization of an incoming EM wave (which, in turn, may bescattered). A formal statement of the voltage equation is [Ken54]:

V = At ER = At[S]ET , (2.45)

where the notation atb ≡ axbx+ayby is for column vectors and superscript t denotes thetranspose. At the beginning of this paragraph, the antenna height A was introducedas the polarization state of a wave transmitted by the receiving antenna towards thetarget. Thus, A is a measurable quantity and is defined on the basis of the radiationpattern of that antenna. The squared absolute value of V ,

PR = |V (A,ER)|2, (2.46)

2.2 - Scattering of electromagnetic waves 17

is described as the power transfer equation.

In [KB86], Kostinski and Boerner dealt with the voltage optimization question, whichis the problem of finding which polarizations of the transmitting and receiving anten-nae maximize the value of P for a target with a known scattering matrix. It must benoted that in (2.45) there is not an inner product because A is not conjugated; hence,for normalized vectors the maximum condition for |V |2 is [Ken52]:

A ∝ E∗R . (2.47)

The physical meaning of this formulation is easily understood if we consider that con-jugation of any PS reverses the sense of rotation of the wave. Hence, the conditionA ∝ E∗

R means that the returned wave is matched to the receiving antenna when itspolarization ellipse is oriented in space identically with the one due to A (radiated bythe receiving antenna when used as a transmitter) and when they have opposite sensesof rotation when both looked at from the same viewpoint.

Kostinski and Boerner provided a general solution for the voltage optimizationproblem which enables one to treat symmetric, asymmetric, monostatic and bistaticcases in an identical manner (further details on the optimal polarization problem andother solutions can be found in [Huy70], [AB89] and [BYXY91]). The approach pro-posed by the two authors has the principal advantage of not requiring diagonalizationof the scattering matrix and therefore the use of change-of-basis formulae. This impor-tant aspect of polarimetry theory will be dealt with in the next paragraph.

When dealing with power measurements, Equation (2.42) may be expressed in termsof Stokes vectors and of the corresponding 4× 4 matrix [K], the so-called Kennaughmatrix, whose elements can be derived from the ones of [S] by means of [vdH81],[BEACM81]:

[K] = [A]∗([S] ⊗ [S]∗)[A]−1, (2.48)

where ⊗ denotes the Kronecker product of the two matrices and [A] is defined as:

[A] =

1 0 0 11 0 0 −10 1 1 00 j −j 0

. (2.49)

For completely polarized waves there is a one-to-one correspondence between the scat-tering matrix [S] and the Kennaugh matrix [K].

In forward scattering cases, [K] must be substituted by the Mueller matrix which isrelated to the Kennaugh one by:

[M] = [C][K] or [K] = [C][M], (2.50)

with:

[C] =

1 0 0 00 1 0 00 0 1 00 0 0 −1

. (2.51)

Before proceeding any further, it is now worthwhile to define some characteristicsof the waves we are considering.


All of the equations introduced in the previous paragraphs refer to monochromaticwaves that cannot be considered as typical real phenomena. Real systems work withpartially polarized waves, which can be expressed, via Fourier integrals, as a superpo-sition of plane monochromatic waves. This is the importance of plane waves that, inthis sense, are the basic elements of all wave problems.

When a partially polarized wave is involved, Equation (2.35) must be replaced by:

g0 ≥ (g21 + g2

2 + g23)

1/2 (2.52)

and the values of the Stokes parameters have to be derived using average estimatesof the measurements, in order to correctly express the statistical variations of the po-larization. Partially polarized waves are generated, typically, after scattering from realtargets which have to be considered as distributed sets of scatterers varying either inspace or in time. They are referred to as non-deterministic or partial or also random scat-terers. In this case it was shown that the average Stokes parameters of the backscat-tered wave are related to those of the illuminating one through an averaged Kennaughmatrix. This matrix can be calculated by averaging the elements of [K] (we will indi-cate this operation using angular brackets 〈〉, that should not be confused with thosedefining the inner product of two vectors), but this process does not preserve the re-lationship with [S]. Hence, the unique connection between the scattering matrix andthe Stokes matrix representation is lost. In such a case, no equivalent scattering matrixexists for the average Kennaugh matrix [Huy70], [vZZE87], [Mot92].

Taking now into account the considerations above, we can show how the receivedpower at the antenna terminal is related to the average Kennaugh matrix. In the mostgeneral case of Figure 2.1, it holds:

PR = |V |2 = |At ER|2 = |〈A|ER〉|2, (2.53)

which, for a monostatic system with fixed polarization, yields:

PR = |〈ET |ER〉|2 = |〈ET |[S]ET 〉|2. (2.54)

Using the Stokes vectors, (2.54) may be written in a similar way:

PR = gtT 〈[K]〉gT , (2.55)

with gT = gT (ψT , χT ) according to (2.36). The value of PR is usually the sum of severalpower measurements to reduce statistical variations due to non-deterministic scatter-ers inside an image resolution cell, so that 〈[K]〉 is the average Kennaugh matrix.

The received power can also be expressed in terms of an area associated with thescatterer, called scattering cross section σ, such that:

σ = limr→∞

(4πr2)

(

PR

PT

)

. (2.56)

At a fixed polarization, i. e., the same polarization of the transmitting and receivingantennae, σ is a function of ψT and χT only (by means of (2.36)). In the most generalcase, one would expect different polarizations and hence: σ = σ(ψR, χR, ψT , χT ).

2.3 - Change-of-basis theory and characteristic polarizations 19

2.3 Change-of-basis theory and characteristic polariza-tions

Let us introduce now some relations that will be extensively used later and let usrefer, for instance, to the wave represented by (2.26). We defined it using a linear basisbut, of course, it would be possible to consider other bases and represent E accordingto them.

What must be stressed is that, regardless of the basis we choose, and therefore of theform in which we represent it, we are dealing with the same wave and there are somerules to follow in order to guarantee proper comparison of measurements taken withdifferent antenna sets. The following requirement must be satisfied while changingbasis: all measurable quantities such as voltage, energy density, etc. must remain invariantunder the change-of-basis.

In general, defining a vector in two different bases requires us to find the relation-ship between its components in the two bases; this relationship is usually expressed bya matrix which maps the components of the vector in one basis to the components inthe other basis.

The fulfillment of the conditions above imposes some constraints on the matriceswhich can be used as change-of-basis matrices: for orthogonal bases, they have to beunitary matrices.

It will be remembered that the adjoint matrix [B]† of a generic complex matrix [B] isdefined as the complex conjugate of its transpose, i. e.:

[B]†def= [B]t∗ (2.57)

and for unitary matrices the following properties hold:5

[U]† ≡ [U]t∗ = [U]−1, (2.58)[U]†[U] = [U][U]† = [I], (2.59)

| det[U]| = 1. (2.60)

In the 2 × 2 complex case, the unitarity requirement imposes four constraints oneight parameters. Hence, [U] is, in general, a function of four variables. In radar po-larimetry it was shown that an appropriate calibration of the system further leads tothe following condition:

det[U] = 1. (2.61)

It is then possible to reduce the number of parameters so that, in terms of the complexpolarization ratio, [U] can be expressed by:

[U] =1√

1 + ρρ∗

[

ejδh −ρ∗e−jδh

ρejδh e−jδh

]

. (2.62)

Let us consider Equation (2.44) and see more in depth how the scattering matrixbehaves under change-of-basis transformations, starting from the forward scattering.

5Equations (2.58) and (2.59) are necessary and sufficient conditions for unitarity whereas (2.60) isonly necessary (but not sufficient).


In such a case, the vectors ER and ET will undergo the transformations:

ER = [U]E′R , (2.63)

ET = [U]E′T , (2.64)

where the primes indicate quantities expressed in the new polarization basis.

What we require is an obvious physical invariance, namely, that the measured volt-age at the receiving antenna terminals remains unchanged under the change-of-basistransformation and that the scattering matrix in (2.44) connects the same physical po-larization states as in the old basis. One then obtains:

ER = [S]ET ⇒ (2.65)[U]E′

R = [S][U]E′T ⇒ (2.66)

E′R = [U]−1[S][U]E′

T , (2.67)

leading to the unitary similarity transformation for the scattering matrix [S]:

[S′] = [U]−1[S][U], (2.68)

which expresses the invariance of the properties of the target operator [S] following thechange-of-basis.

Significant differences characterize the scattering equation in the monostatic case.Also for backscattering, the two polarization states (of the transmitted and of the re-ceived wave) can be referred to using the same reference system and expressed in thesame basis but, obviously, correspond to waves propagating in opposite directions. Itis then important to correctly define their handedness. This is possible by means of thedirectional Jones vectors. In a linear basis, keeping in mind that E± = E∗

∓ (see (2.29)),Equation (2.44) must be rewritten as:

ER,− = [S]ET,+ = [S]E∗T,− . (2.69)

According to this new expression (we will assume that, for a correct comparison, thewaves propagate in the same direction and for simplicity we will ignore the subscript“−”), the change-of-basis relation for the scattering matrix becomes:

ER = [S]E∗T ⇒ (2.70)

[U]E′R = [S]([U]E′

T )∗ ⇒ (2.71)E′

R = [U]−1[S][U]∗E′∗T , (2.72)

which again leads to:[S′] = [U]−1[S][U]∗ = [U]†[S][U]∗. (2.73)

Simply changing the name of [U]∗ and letting [U]∗ = [U1] , it becomes evident thatthe representation of the scattering matrix [S′] in the new polarization basis is obtainedby a unitary congruence transformation of the original scattering matrix [S]:

[S′] = [U1]t[S][U1] . (2.74)

This relation plays a central role in radar theory since most of systems have a monos-tatic configuration.

2.3 - Change-of-basis theory and characteristic polarizations 21

The use of the change-of-basis formulae permits the expression of the scatteringmatrix in diagonalized forms; hence, one has access to particular representations ofthe interactions where the backscattered power is concentrated only in some of thecomponents of [S].

Let us consider the matrix transformation from the (h, v) linear basis into a genericone with basis vector (e1, e2):

[S]=1

N

[

1 ρ−ρ∗ 1

] [

Shh Shv

Svh Svv

] [

1 −ρ∗ρ 1

]

=

=1

N

[

1 ρ−ρ∗ 1

] [

Shh + ρShv −ρ∗Shh + Shv

Svh + ρSvv −ρ∗Svh + Svv

]

=

=1

N

[

Shh + ρShv + ρSvh + ρ2Svv −ρ∗Shh + Shv − ρρ∗Svh + ρSvv

−ρ∗Shh + Svh − ρρ∗Shv + ρSvv ρ∗ρ∗Shh − ρ∗Shv − ρ∗Svh + Svv

]

,

(2.75)

where:N = 1 + ρρ∗ (2.76)

(we will no longer add the numerical subscripts to the [S] matrix elements, adoptingthe broadly used convention of considering the first subscript related to the receivingsystem and the second one to the transmitting system).

As reported in [Ken54], the two cross-polar nulls, which define the polarization vec-tors whose reflected wave has zero cross-polar components, coincide with the co-polarmaxima. To obtain them, one has to impose one of the following zeroing conditionson the cross-polar terms of (2.75) (due to reciprocity, in the monostatic case they areequivalent):

−ρ∗Shh + Shv − ρρ∗Svh + ρSvv = 0 (2.77)−ρ∗Shh + Svh − ρρ∗Shv + ρSvv = 0 (2.78)

Considering the former, taking its complex conjugate and subtracting the latter:{

(ρ∗Shh − Shv + ρρ∗Svh − ρSvv)S∗hv = 0

(ρS∗hh − S∗

hv + ρρ∗S∗vh − ρ∗S∗

vv)Shv = 0(2.79)

results in [BEACM81]:

ρ∗ShhS∗hv + ρ∗S∗

vvShv − ρSvvS∗hv − ρS∗

hhShv = 0 ⇒ (2.80)ρ∗(ShhS

∗hv + S∗

vvShv) = ρ(S∗hhShv + SvvS

∗hv) ⇒ ρ∗A∗ = ρA, (2.81)

being:A = S∗

hhShv + SvvS∗hv. (2.82)

Now, taking again (2.77) and multiplying it by A∗:

ρ∗A∗Shh + ρρ∗A∗Shv − SvhA∗ − ρA∗Svv = 0 ⇒ (2.83)

ρ2AShv + ρ(AShh − A∗Svv) − SvhA∗ = 0 ⇒ (2.84)

ρ2AShv + ρ(S∗hhShvShh +

+SvvS∗hvShh − ShhS

∗hvSvv − S∗

vvShvSvv) − SvhA∗ = 0 ⇒ (2.85)

ρ2A+ ρ(|Shh|2 − |Svv|2) − A∗ = 0. (2.86)


Equation (2.86) has two possible complex solutions in terms of ρ:

ρ(p, q)1, 2 =

|Svv|2 − |Shh|2 ±√

(|Shh|2 − |Svv|2)2 + 4 |A|2

2A(2.87)

that lead to two diagonalized [S] matrices:

ρ(p, q)1 ⇒ [S](p1, q1) =

[

p 1 00 q1

]

, (2.88)

ρ(p, q)2 ⇒ [S](p2, q2) =

[

p 2 00 q2

]

, (2.89)

where the values of the two pairs of eigenvalues, are obtained by substituting the twosolutions found with (2.87) into (2.75).

It can also be shown that the cross-polar nulls are always mutually orthogonal, in-deed:

(ρ(p, q)1 ) · (ρ(p, q)

2 )∗ =

−B +√

B2 + 4 |A|2

2A

−B −√

B2 + 4 |A|2

2A

∗

=

=B2 −B2 − 4 |A|2

4AA∗= −1, (2.90)

with:B = |Shh|2 − |Svv|2 . (2.91)

The other group of characteristic polarizations is represented by the [S] matrix co-polar nulls. As in the previous case, they can be calculated by setting both the co-polarterms of (2.75) to zero:

Shh + 2ρShv + ρ2Svv = 0 (2.92)Svv − 2ρ∗Shv + (ρ∗)2Shh = 0. (2.93)

The solutions of these two equations lead to two pairs of values for ρ, respectively(ρ

(x)1 , ρ

(x)2 ) and (ρ

(y)1 , ρ

(y)2 ), and to two pairs of matrices which differ only in their phase

terms. Indeed one gets:

ρ(x)1 ⇒ [S]x1

=

[

0 x1

x1 a1

]

, (2.94)

ρ(x)2 ⇒ [S]x2

=

[

0 x2

x2 a2

]

, (2.95)

ρ(y)1 ⇒ [S]y1

=

[

b1 y1

y1 0

]

, (2.96)

ρ(y)2 ⇒ [S]y2

=

[

b2 y2

y2 0

]

, (2.97)

with:|x1| = |x2| = |y1| = |y2| (2.98)

2.4 - Scattering vectors and second order matrices 23

and|a1| = |a2| = |b1| = |b2|. (2.99)

It is important to note here that a more formal approach to the diagonalization ofthe scattering matrix is possible (for instance, by means of Takagi’s theorem [HJ85])which prevents some limits of the equations and derivations above. Indeed, these failwhen diagonalizing some special scattering matrices. It is nevertheless possible to usethem since in the majority of the cases, as it has been proven with experimental data,they do operate correctly.

In summary, by measuring the target scattering matrix at a fixed observation geome-try and frequency, the invariant target parameters may be calculated and the resultingtarget information used to improve signal-to-clutter ratio or to provide the basis fortarget identification using pattern recognition techniques. These properties will be ap-plied to real data in order to test their capability to describe natural scenes.

2.4 Scattering vectors and second order matrices

So far our attention has been mainly directed toward the scattering matrix. However,other representations of scattering phenomena, based on second order matrices, proveto be useful since they can better deal with the real case of partially polarized waves.It is worthwhile to introduce them before considering, in the next chapter, practicalapplications of SAR polarimetry.

Let us begin by rewriting the scattering matrix as a scattering vector, by means of thefollowing relation [Clo86], [CP96]:

k(4) =

k0

k1

k2

k3

def=

1

2

Trace([S][Ψ0])Trace([S][Ψ1])Trace([S][Ψ2])Trace([S][Ψ3])

, (2.100)

where Trace[B] is the sum of the diagonal elements of whatever matrix [B] and Ψ =([Ψ0], [Ψ1], [Ψ2], [Ψ3]) is a set of 2 × 2 complex matrices that form an orthogonal basis.

Among the basis sets used in the literature the most important ones are the follow-ing:

ΨL =

(

2

[

1 00 0

]

, 2

[

0 10 0

]

, 2

[

0 01 0

]

, 2

[

0 00 1

])

(2.101)

and

ΨP =(√

2 [σ0],√

2 [σ1],√

2 [σ2],√

2 [σ3])

=

=

(√2

[

1 00 1

]

,√

2

[

1 00 −1

]

,√

2

[

0 11 0

]

,√

2

[

0 −jj 0

])

. (2.102)

The former is a straightforward lexicographic ordering of the elements of [S], whilethe latter is based on the Pauli matrices.


Performing the vectorization of [S] using (2.100), two different scattering vectors canbe derived from the bases above:

k(4)L =

Shh

Shv

Svh

Svv

, (2.103)

k(4)P =1√2

Shh + Svv

Shh − Svv

Shv + Svh

j(Shv − Svh)

. (2.104)

Particular attention must be paid to the Pauli matrix basis: as will be shown later, itselements are related to elementary scattering mechanisms so that the scattering vectorscan be immediately associated with concrete physical phenomena. For this reason it isalso interesting to express the matrix [S] in terms of the elements of the Pauli scatteringvector:

[S] =

[

k0 + k1 k2 − jk3

k2 + jk3 k0 − k1

]

. (2.105)

The multiplication factors 2 and√

2, which appear in (2.101) and (2.102), are neededin order to keep the norm of the scattering vector, which is equal to the total scatteredpower, independent of the chosen matrix basis Ψ; in fact:

‖k(4)‖2 = 〈k(4)|k(4)〉 = Span([S]) =

= Trace([S][S]†) = Trace([S]†[S]) =

= |Shh|2 + |Shv|2 + |Svh|2 + |Svv|2. (2.106)

Again, as seen for the change-of-basis of the PSs (see page 19), the transformationfrom the lexicographic into the Pauli representation of the scattering vectors can beobtained by means of a relation of the type:

k(4)P = [D(4)]k(4)L, (2.107)

where [D(4)] is a unitary matrix defined as [CP96]:

[D(4)] =1√2[A] (2.108)

with the matrix [A] given by (2.49).

We anticipated on page 18 that real systems involve scatterers situated in dynamicenvironments and subject to space and/or time variations (non-deterministic scatter-ers). This causes the EM waves to be partially polarized and prevents the scatteringprocess from being described by a single matrix [S] only and hence by a single scatter-ing vector.

In order to study such phenomena, it is useful to introduce some new matrices,starting from the covariance matrix, obtained by performing the outer product of the

2.4 - Scattering vectors and second order matrices 25

lexicographic scattering vectors:

[C(4)]def= 〈k(4)L k

†

(4)L〉 =

=

〈|Shh|2〉〈ShhS∗hv〉〈ShhS

∗vh〉〈ShhS

∗vv〉

〈ShvS∗hh〉〈|Shv|2〉〈ShvS

∗vh〉〈ShvS

∗vv〉

〈SvhS∗hh〉〈SvhS

∗hv〉〈|Svh|2〉〈SvhS

∗vv〉

〈SvvS∗hh〉〈SvvS

∗hv〉〈SvvS

∗vh〉〈|Svv|2〉

. (2.109)

A similar expression can be derived by means of the Pauli scattering vectors, whichbrings to the coherency matrix [T(4)]:

[T(4)]def= 〈k(4)P k

†

(4)P 〉. (2.110)

Of course, the ensemble averaging becomes redundant for deterministic scatterers;both matrices are by definition Hermitian positive semidefinite.

The relation between [C(4)] and [T(4)] follows from straightforward mathematics:

[T(4)] = 〈k(4)P k†

(4)P 〉 = 〈[D(4)]k(4)L k†

(4)L[D(4)]†〉 =

= [D(4)]〈k(4)L k†

(4)L〉[D(4)]† = [D(4)][C(4)][D(4)]

†. (2.111)

We know that in the backscattering case the reciprocity theorem constrains scatter-ing matrices to be symmetric, i. e. Shv = Svh ; in such a case the dimensions of the scat-tering vectors, and hence of the coherency and covariance matrices, may be reduced asfollows:

k(3)L = [Q]k(4)L =

Shh√2Shv

Svv

, (2.112)

where:

[Q] =

1 0 0 0

0 1/√

2 1/√

2 00 0 0 1

(2.113)

(with [Q][Q]t = [I(3)]), and in a similar way:

k(3)P =1√2

Shh + Svv

Shh − Svv

2Shv

. (2.114)

This leads to covariance and coherency matrices of smaller dimensions, 3 × 3, definedas:

[C(3)]def= 〈k(3)L k

†

(3)L〉 =

=

〈|Shh|2〉√

2〈ShhS∗hv〉〈ShhS

∗vv〉√

2〈ShvS∗hh〉 2〈|Shv|2〉

√2〈ShvS

∗vv〉

〈SvvS∗hh〉

√2〈SvvS

∗hv〉〈|Svv|2〉

(2.115)


and

[T(3)]def= 〈k(3)P k

†

(3)P 〉 =

=1

2

〈|Shh + Svv|2〉〈(Shh + Svv)(Shh − Svv)∗〉 2〈S∗

hv(Shh + Svv)〉〈(Shh + Svv)

∗(Shh − Svv)〉〈|Shh − Svv|2〉 2〈S∗hv(Shh − Svv)〉

2〈Shv(Shh + Svv)∗〉 2〈Shv(Shh − Svv)

∗〉 4〈|Shv|2〉

.

(2.116)

The transformation from one representation to the other is again obtained using aunitary matrix, so that:

k(3)P = [D(3)]k(3)L (2.117)

and[T(3)] = [D(3)][C(3)][D(3)]

†, (2.118)

with:

[D(3)] =1√2

1 0 11 0 −1

0√

2 0

. (2.119)

2.5 Target decomposition theorems

By means of the new representations provided by the scattering vectors and the covari-ance and coherency matrices, it is now possible to discuss the target decomposition (TD)theorems. These methods provide a physical interpretation of the scattered signals, thatis achieved by considering them as superpositions of several contributions and tryingto recognize each of them. This approach was first outlined by Chandrasekhar for lightscattering by small anisotropic particles and later applied to polarized MW by Huynen[Huy70].

Following Huynen’s fundamental contribution, several decomposition techniqueswere proposed, some of them attempting to overcome some limitations of Huynen’smethod.

According to [CP96], three primary classes of such decompositions may be defined:

• coherent theorems, that decompose the [S] matrix into the sum of elementary ma-trices;

• Huynen type decompositions, which attempt to extract a single scattering matrixfrom the average Kennaugh matrix;

• eigenvector decompositions of the coherency or covariance matrices.

Coherent decomposition theorems use [S] matrices, and their underlying principleis to consider a generic scattering matrix as a linear combination of several others, eachdefining a simple deterministic scatterer. Unlike the above mentioned decompositionsof power reflection matrices, decompositions of the scattering matrix are particularlysuited for cases where the scattering is due to few dominant scattering centres.

2.5 - Target decomposition theorems 27

For example, using the Pauli matrices it is possible to write:

[S] =

[

a+ b c− jdc+ jd a− b

]

=

= a

[

1 00 1

]

+ b

[

1 00 −1

]

+ c

[

0 11 0

]

+ d

[

0 −jj 0

]

, (2.120)

where a, b, c and d are all complex and correspond to the elements of the Pauli scatter-ing vector (see Equation (2.105)).

According to the hypotheses that have been made, this decomposition yields the co-herent sum of four scattering mechanisms: the first being single scattering from a planesurface, the second and third being diplane scattering from corner reflectors with a rel-ative orientation of 45◦ and the final element being all the antisymmetric componentsof the matrix [S] (which corresponds to a scatterer that transforms every incident po-larization into its orthogonal state). As it causes [S] to be non-symmetric, the last termdisappears, of course, in reciprocal backscattering cases.

Another example of a coherent TD method was presented by Krogager [Kro92],[Kro93]. His approach was based on the observation that any complex, symmetricscattering matrix can be decomposed into three components, as if the scattering weredue to a sphere, a diplane and a right- or left-rotating helix. This can be shown by ma-nipulating the real and imaginary parts of the elements of [S], resulting in the followingformulation:

[S] = ks[S]sphere + ejφ(kd[S]diplane + kh[S]helix), (2.121)

where ks, kd and kh are real quantities and φ represents a relative phase which is equalto the displacement of the diplane and the helix relative to the sphere.6

The sphere, diplane and helix (henceforth denoted SDH) terms are related to mea-surable and familiar quantities, in that ks and kd can be measured directly using circularpolarizations. In relation to target imaging, this decomposition provides a means forrepresenting the target return in three different images, each characterizing differentscattering mechanisms. The advantage is that for those cases where one of the types ofscattering is predominant, the image corresponding to that type will be the only onecontaining significant response from the resolution cell in question. A resulting colourcomposite image would behave in this way, providing a valid classification systembased on the relative contribution of the three mechanisms. This is particularly usefulfor practical applications, because the radar scattering can often be ascribed to one ormore individual scatterers within each resolution cell. These individual contributionsare typically due to major surfaces and to two- or three-sided corners reflectors (dihe-dral and trihedral). The role of the helix in the above decomposition becomes apparentfrom the fact that helix-like scattering may be produced by two or more dihedrals, de-pending on their relative orientation angles and displacements. On the other hand,helices as such are rarely found in practice. This means that, for resolution cells wherea significant amount of helix-like scattering is found, one may conclude that the scat-tering from those resolution cells is mainly due to two or more even-bounce reflectionmechanisms. For diagnostic purposes such information is obviously useful. Likewise

6In Appendix B are reported the exact expressions of the k coefficients as well as details on the fol-lowing example of coherent decomposition, the Cameron one.


it is very useful to be able to determine whether the dominating scattering from a res-olution cell is due to a single bounce mechanism or to a double bounce mechanism[vZZE87], [vZ89].

The approach proposed by Cameron [CL90], [CYL96] may be considered as a moregeneralized example of the “model fitting” seen with the SDH decomposition. Givena generic matrix [S] (not only in the monostatic case), one can characterize it by itstendency of being more or less symmetric according to the reciprocity rule and splitit into two terms representing reciprocal and non-reciprocal scattering mechanisms.Using the k(4)L vector corresponding to [S] this decomposition is expressed by:

k(4)L = krec + knon−rec . (2.122)

In turn, the reciprocal term (the one usually available after calibrating monostatic radardata) represents a target which is more or less symmetric with respect to an axis in theplane orthogonal to the radar line-of-sight and again a distinction can be made betweenthe most dominant and the least dominant symmetric target components,7 that is:

krec = kmaxsym + kmin

sym . (2.123)

Hence, the decomposition follows the scheme:

[S] ��*

[S]rec

HHHj [S]non−rec

��*

[S]maxsym

HHHj [S]min

sym

The degrees of reciprocity and symmetry are evaluated in terms of projection anglesof the scattering vectors onto the corresponding subspace and subsets via proper pro-jection operators. In a similar way, one may compare an arbitrary scattering matrix toa model scattering matrix. In [CL90], this principle was further pursued leading to amatrix classification scheme able to assign [S] to one of eleven classes according to thedegree of reciprocity, symmetry and resemblance to a set of model matrices.

In contrast to coherent methods, Huynen decomposition [Huy70] acts on the Ken-naugh matrix and is based on two key principles: to consider distributed targets forwhich 〈[K]〉 has no single equivalent scattering matrix; to try and extract from thisaverage a “single” Kennaugh matrix (and hence a corresponding [S] matrix) and a “N-targets” (i. e., random) contribution. This idea was expressed in mathematical terms inthe following way:

〈[K]〉 = [Ks] + 〈[KN ]〉. (2.124)

One of the limits of this approach is that, in general, it does not provide the invarianceof the residue N-matrix under all possible unitary transformations [Clo92].

In fact, Huynen decomposition can also be performed using an equivalent coherencymatrix representation; as this other form can be used to generate a diagonal coherency

7One should note the difference in the use of the word “symmetry” when referred to scatteringmatrices and to targets. Indeed, according to the Cameron decomposition, scattering matrices whichare symmetric due to the reciprocity constraint may refer to targets which are geometrically more or lesssymmetric in the plane orthogonal to the radar line-of-sight (in the case of a helix, a symmetric scatteringmatrix describes a target which is not geometrically symmetric).

2.5 - Target decomposition theorems 29

matrix, which may be physically interpreted as statistical independence between a setof target vectors, such an approach yields a general decomposition into independentscattering processes. The general expression of this diagonalization is the following:

[T(3)] = [U(3)][Λ][U(3)]−1, (2.125)

where:

[Λ] =

λ1 0 00 λ2 00 0 λ3

(2.126)

is the diagonal matrix with elements the real non-negative eigenvalues, 0 ≤ λ1 ≤ λ2 ≤λ3, of [T(3)] (the coherency matrix is, indeed, Hermitian positive semidefinite) and

[U(3)] =[

e1 e2 e3

]

(2.127)

is the unitary eigenvector matrix built using as columns the eigenvectors of [T(3)]. Theeigenvectors are orthogonal since the coherency matrix is Hermitian.

By means of the diagonalization of [T(3)], which is in general of rank 3, the interac-tion is represented by the non-coherent sum (see the definition (2.116) of [T(3)]) of threeindependent (in the sense that they are connected to orthogonal vectors) coherencymatrices [T(3)i], each weighted by its eigenvalue:

[T(3)] = [U(3)][Λ][U(3)]−1 =

3∑

i=1

λi[T(3)i] =3

∑

i=1

λi eie†i . (2.128)

For each [T(3)i] matrix it is possible to define a corresponding scattering matrix (i. e.,a deterministic scattering contribution), while the choice of the eigenvectors for build-ing the unitary matrix [U(3)] always guarantees an orthogonal basis set. The corre-spondence between the resulting coherency matrices and their respective scatteringmatrices may be referred as well to the relative eigenvectors so that also these can beused to represent scattering mechanisms as the [S] matrices.

3 SAR polarimetry

3.1 Basics

In this section, SAR systems for polarimetric applications will be introduced. At first,a general review of the SAR sensors presently operating will be made, of the single-polarization systems as well as of the multi-polarimetric ones. Then, only polarimetricSARs will be considered and some aspects of their practical applications will be ana-lyzed. This should provide a first insight on the potentials of these systems.

Theoretical questions which derive from the complexity of the topic, will be dis-cussed later on in this chapter.

3.1.1 SAR sensors

As stated before, synthetic aperture radars present important advantages with respectto conventional ones. Thus, since the beginning of the eighties, a great deal of workhas been performed using these systems. At present, there are several SAR sensorsin operation: single-polarization systems as well as multi-polarimetric ones. Note-worthy examples in the first group are the satellite mounted SAR systems: ERS-2(by ESA, operating in C-band), JERS-1 (Japanese, L-band) and RADARSAT-1 (Cana-dian, C-band) and various airborne systems, namely: TOP-SAR (NASA/JPL, C- andL-band), IFSARE (Environmental Research Institute of Michigan, X-band), C/X-SAR(CCRS, Canada, X- and C-band), EMI-SAR (Danish Defence Research Establishment,C-band), E-SAR (DLR, X-, C-, L- and P-band), DoSAR (Dornier, Germany, X- andC-band), AeS-1 (AeroSensing, Germany, X-band), RAMSES (Onera, France, X-band),AER-II (FGAN, Germany, X-band) and ESR (DERA, UK, X-band). In the second groupare included the airborne radars: AIRSAR (by NASA/JPL, which operates in fully po-larimetric mode in C-, L- and P-band and which has to be mentioned as the forerunnerof all polarimetric SAR), EMI-SAR (DDRE, Denmark, C- and L-band), E-SAR (DLR, L-and P-band) and NABC/ERIM SAR (ERIM, USA, X-, C- and L-band), and the ASARC-band sensor mounted on the ESA ENVISAT satellite. To both groups belongs theShuttle Imaging Radar SIR-C/X-SAR, which was flown twice in April and October1994 and later, in February 2000, with a second antenna mounted on a 60 m boom forthe Shuttle Radar Topography Mission (SRTM): part of the data of the missions wereacquired in fully polarimetric mode in C- and L-band.

The main advantages of SAR systems arise from their imaging geometry, which re-quires a moving antenna (see Figure 3.1). Scene imaging with a moving device permits

31

32 Chapter 3 - SAR polarimetry

� � � ��

� � � ��

look angle, θ

R

swath

nadir track

antenna track

SAR antenna

footprint

h

� � �&�� ) +-,�� . ! �� '��&. ��

the simulation of antenna dimensions much larger then the real ones and makes reso-lution along the flight-path independent of the sensor-target distance.

3.1.2 Early experimental results

Some of the earliest results obtained with SAR polarimetric systems were presentedin a series of related papers by Zebker et al. [ZvZH87], van Zyl et al. [vZZE87] andEvans et al. [EFvZZ88], following data acquisition campaigns of the NASA/JPL high-resolution airborne imaging polarimetric SAR (further details on this apparatus, oper-ating mainly in L-band and later extended to C- and P-band, can be found in [ZvZ91]).In particular, van Zyl et al. [vZZE87] reported experimental proofs of some basic scat-tering models: from a large, smooth dielectric surface, from a rough surface and froma dihedral corner reflector.

In [vZZE87], the scattering cross section σ = σ(ψT , χT ) introduced with (2.56) iscalled the polarization signature of the scatterer and is widely used as it provides a re-ally useful graphical representation of the behaviour of different targets. Indeed, theinformation contained in σ(ψT , χT ) may be displayed as three-dimensional figures ex-pressing the intensity of the backscattering from a target for varying ellipticity χT andellipse orientation ψT of the transmitted wave.

The polarization signature, through its plots, has proven to be a valid tool in pat-

3.1 - Basics 33

tern recognition. For example, a dihedral corner reflector, where the transmitted wavesuffers two reflections, has a polarization signature which differs significantly fromthat of a wave which is scattered by a rough surface. With respect to these models,backscatter from ocean exhibits a polarization signature similar to that predicted bythe one of scattering by a slightly rough dielectric surface. Urban regions exhibit thecharacteristics expected from dihedral corner reflectors and their polarization signa-ture is quite different from that of the ocean or of the single-reflection slightly roughmodel [EFvZZ88], [ZvZ91].

Rough surface scattering presents an interesting behaviour: in this case, the three-dimensional plot of σ(ψT , χT ) has minima which never go to null (as happens for othermodel scatterers). Hence, a “pedestal” of power returns always greater than zero ap-pears in the polarization signature of rough surfaces. As pointed out in [vZZE87], thismeans that there is some portion of the return which cannot be nulled by changingthe polarization of the transmitting and receiving antennae. In polarimetric observa-tions, the presence of a pedestal in the signature means that the individual measuredmatrices used to calculate the signature were not identical. The more different the in-dividual matrices, the higher the resulting pedestal. In radar images, this variationof scattering properties results from various different effects which all cause the mea-sured scattering, and hence Kennaugh matrices of adjacent resolution elements, to beslightly different. The most obvious cause of such a variation is when adjacent pix-els really contain different types of scatterers. Other causes include diffuse (that is,multiple) scattering and the presence of noise [vZZ90].

Based on these results, an unsupervised classification1 algorithm able to distinguishbetween scattering with even number of reflections, odd number of reflections anddiffuse scattering, was tested with good results and presented in a paper by van Zylin 1989 [vZ89]. It was shown, for example, that information about the health of forestscan be retrieved without ancillary data, just using this broad class division.

Unfortunately, polarization signatures are not unique. For instance, the polarizationsignature of a dielectric sphere has the same form as that of a smooth dielectric surfaceat normal incidence. It is also conceivable that different combinations of various scat-tering mechanisms may yield the same resulting σ(ψR, χR, ψT , χT ). However, under amore general point of view, it was evident that different polarizations (i. e., differenttransmitting and receiving settings) would provide different information according tothe characteristics of the observed scene.

Further aspects of pattern recognition and image interpretation methods will be an-alyzed in following sections of this work as this area of study frequently overlaps withthose of the inversion methods and, when used for these purposes, of the interferomet-ric applications.

As stated before, an important characteristic of radar devices is that they permitmeasurements of the phase of the signal at a receiving antenna. In this way, besidesstudies on the scattered power (as the series reported above), it is possible to analyzethe phase of both the orthogonal components of an EM wave.

An interesting example of these possibilities was reported in a paper by Ulaby et al.in 1987 [UHD+87]. Again the data used were those of the JPL L-band SAR, but the

1A definition of unsupervised classification may be found in Paragraph 4.2.


attention was there devoted to the statistical behaviour of the phase difference betweenthe hh-polarized and the vv-polarized backscattered signals:

φhhvv = φhh − φvv. (3.1)

Measurements were performed over agricultural test sites. While for the over-whelming majority of cases the φhhvv distribution was symmetrical and centered arounda zero mean value, a different pattern was observed for corn fields: the mean phasedifference was different from zero and increased with increasing look angle θ (see Fig-ure 3.1). The explanation proposed for this variation was that the corn canopy, mostof whose mass is contained in its vertical stalks, acts like uniaxial crystal characterizedby different velocities of propagation for waves with horizontal and vertical polariza-tion. Hence, assuming further that the observed backscatter was contributed by di-rect backscatter from plants, direct backscatter from soil and double-bounce reflectionon soil and plants, a good agreement was found between measurements and model-derived expectation values.

About the expected distribution of φhhvv, a valid model was later proposed by Sara-bandi [Sar92]. According to his general hypotheses, the probability density functionsof the phase differences of the scattering matrix elements were derived from the av-erage Kennaugh, assuming that the elements of [S] are jointly Gaussian. As a result,the expressions defined for the co- and cross-polarized phase differences were similarin form and could be considered analogous to the Gaussian distribution for periodicrandom variables.

It must be stressed that, regardless of the interesting features related to this kind ofanalysis, studies on the phase behaviour of the scattered signals have not been fullyexploited and can be considered as a promising subject for further investigations.

3.1.3 Calibration techniques

A completely different research topic on SAR polarimetric data arises from the cali-bration problem [ZvZ91], [Fre92] (the latter also for a particularly rich bibliography onthis subject).

A fully polarimetric radar measures the full scattering matrix for every resolutionelement in an image. The increase of information with respect to “scalar” radar comesat a price, not only in the increase in complexity of the design and in the cost of buildingthe radar system and processing the data, but also in the amount of effort needed tocalibrate the data. Moreover, calibration is specific and peculiar to each apparatus sinceeach has its own calibration needs.

In [vZ90], a technique was described which used the theoretical result that, for nat-ural targets with azimuthal symmetry, the co- and cross-polarized components of thescattering matrix are uncorrelated. Representing this in mathematical form, this meansthat:

〈S∗hhShv〉 = 0 (3.2)

and〈S∗

vvSvh〉 = 0 , (3.3)

3.2 - Interactions with the Earth surface 35

where, again, 〈〉 represents the ensemble average over an extended area (i. e., overseveral resolution elements).

The tests reported in [vZ90] confirmed the validity of the assumptions (3.2) and (3.3)for several terrain types, including desert surfaces, ocean surfaces, grasslands and seaice, and only showed departures for heavily vegetated areas. The main advantage ofthis calibration technique was that it does not require any external calibration deviceto be deployed in the observed area before imaging. Besides this, the whole calibrationprocess resulted in being separable in several independent stages and the cross-talkcalibration to be a well defined part of it.

The already cited paper by Freeman [Fre92] is, instead, a more general survey oncalibration problem, with a wide descriptive purpose; along with other techniquesavailable for polarimetric data, examples are provided also for methods for calibratinginterferometric SAR data.

3.2 Interactions with the Earth surface

When describing the first experimental applications of polarimetric SAR, we referredseveral times to “smooth” and “rough” surfaces. Let us now see more carefully whatwe meant with these terms and how the roughness of a surface actually influences thescattering of an EM wave.

In general, all natural surfaces may be considered as rough but roughness is not anintrinsic property of the scattering surfaces since it depends on the properties of theincident wave. Indeed, both the frequency and the incidence angle of the transmittedwave determine how rough or smooth a certain surface appears to be.

Considering constant wavelength and local incidence angle, the interaction of anEM wave with surfaces of different roughness may be simplified as follows: the rougherthe surface, the more diffuse the scattering or the smoother the surface, the more di-rectional the scattering. The case of an ideal smooth boundary surface between twodielectric media, for instance the air and a homogeneous soil, was explained by Fres-nel who also determined the reflection coefficient of the horizontal and vertical com-ponents of an incoming wave as:

Γh(θ) =µ cos θ −

√

µε− sin2 θ

µ cos θ +√

µε− sin2 θ, (3.4)

Γv(θ) =ε cos θ −

√

µε− sin2 θ

ε cos θ +√

µε− sin2 θ, (3.5)

where µ and ε are, respectively, the dielectric constant and the magnetic permeabilityof the soil and θ the wave incidence angle.

In natural environments, the hypothesis of smooth boundary surface is hardly metso that also the scattering presents different characteristics. The backscattered EMwave from a natural surface consists of two components: a reflected, coherent partand a diffused, incoherent one (see Figure 3.2). The former resembles the behaviourof specular reflection on a smooth surface and thus, in the case of a monostatic radar,


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gives no scatter return; the latter is a diffuse scatter and distributes the scattered powerin all directions. As the surface becomes rougher, the coherent component becomesnegligible and the incoherent one consists of only diffuse scattering.

From an electromagnetic point of view, the definition of a surface as rough or smoothis somewhat arbitrary. Nevertheless, in the literature two main criteria can be found todefine the roughness of a surface: the Rayleigh and the Fraunhofer criterion [UMF82].Considering a plane monochromatic wave transmitted at some angle θ onto a roughsurface as represented in Figure 3.3, one can easily calculate the phase difference ∆φbetween two rays scattered from different points on the surface:

∆φ = 2h2π

λcos θ, (3.6)

where h expresses the standard deviation of the roughness height with respect to areference plane and λ is the wavelength of the incident wave. The Rayleigh criterionstates that if the phase difference ∆φ is less than π/2 radians, i. e., if

h <λ

8 cos θ, (3.7)

then the surface may be considered as smooth. The Fraunhofer criterion is more strin-

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3.3 - Entropy/α analysis 37

gent and requires a ∆φ less than π/8 radians to define a surface as smooth, hence:

h <λ

32 cos θ. (3.8)

3.3 Entropy/α analysis

Among the theoretical methods that have been recently applied to the interpretation ofSAR polarimetric data, the entropy/α analysis seems to be particularly promising andinteresting.

Using the eigenvector-based decomposition of the coherency matrix discussed onpage 29, the relative importance of each scattering mechanism (within a given resolu-tion cell) is expressed by means of its eigenvalues. Indeed, whereas the eigenvectorsdiscriminate the presence of different scattering behaviours, the eigenvalues underlinethe intensity of each mechanism. A quantity that measures the randomness of thesescattering processes is the polarimetric scattering entropy2 H defined as follows:

H =3

∑

i=1

−Pi log3 Pi , (3.9)

where:Pi =

λi

3∑

l=1

λl

. (3.10)

Pi represents the “appearance probability” of each contribution. H ranges from 0 to 1:

• an entropy equal to 0 corresponds to a deterministic scattering process ([T(3)] hasonly one non-vanishing eigenvalue);

• an entropy equal to 1 indicates a degenerated eigenvalue spectrum, typical ofrandom noise processes ([T(3)] has three identical eigenvalues).

To estimate the relative importance of the different scattering mechanisms, a secondpolarimetric indicator has been introduced, the polarimetric anisotropy A:

A =λ2 − λ3

λ2 + λ3

. (3.11)

When H tends to 1 (i. e., when: λ1 ' λ2 ' λ3) or to 0 (λ2 ' λ3 ' 0), A gives nofurther information except for low and medium H (λ1 > λ2, λ3), whereas the entropysays nothing about the relationship between the two minor eigenvalues; the anisotropysupplies this information. Thus, a medium entropy means that more than a singlescattering mechanism contributes to the back-scattered signal, but it is not clear howmany additional mechanisms are present (one or two). A high A states that only the

2The scattering entropy formulation was first introduced to describe the depolarization degree of anEM wave (see [BW85]).


second scattering mechanism is important, whereas a low A indicates a remarkablecontribution of the third one as well.

It is important to note that the set of orthogonal components represented by thethree eigenvectors has no direct physical significance in terms of real scattering mech-anism. In order to provide a physical interpretation, one needs to relate the degrees offreedom of the mathematical problem to the observed scattering process [Pap99]. Oneshould first observe that, in the general bistatic case, the 4×4 covariance and coherencymatrices contain sixteen independent parameters, namely four real power values andsix complex cross-correlations; in the backscattering case the matrices reduce to 3 × 3ones and contain nine independent parameters. On page 16, we have seen that a co-herent matrix [S] has instead seven independent parameters in the bistatic case andfive in the reciprocal monostatic one. It is then evident that some limits exist for a sin-gle scattering matrix to represent a partial scatterer since the latter has, at least, fourmore degrees of freedom. The question is then how to interpret these extra degrees offreedom which the coherency (or covariance) matrix seems to take into account.

Let us reconsider the backscattering case and the k vector derived in the Pauli matrixbasis as expressed by Equation (2.114).

The terms that there appear are simply the three complex elements of the [S] ma-trix: Shh, Svv and Shv. This implies that the scattering vector is characterized by sixindependent parameters.

Further, k(3)P can be written as:

k(3)P = ‖k(3)P‖w, (3.12)

introducing the normalized vector w which has only five degrees of freedom becauseof the normalization constraint. Since each generic k vector is associated with an [S]matrix, and hence to a scattering mechanism, so is each w vector. The five independentparameters may be associated with five angles, so that:

w =

cosα exp(−jφ)sinα cosβ exp(−jδ)sinα sinβ exp(−jγ)

. (3.13)

A change ∆α and ∆β of the two angles α and β causes a transformation of w de-scribed by two rotation matrices:

w′ =

cos ∆α − sin ∆α 0sin ∆α cos ∆α 0

0 0 1

w (3.14)

and

w′ =

1 0 00 cos ∆β − sin ∆β0 sin ∆β cos ∆β

w. (3.15)

This observation leads to the scattering vectors reduction theorem: it is always possibleto reduce an arbitrary scattering mechanism, represented by a complex unitary vector

3.3 - Entropy/α analysis 39

α = 0

Isotropicsurfaces[

1 00 1

]

Anisotropic surfaces� -

α = π4

Dipoles

[

1 00 0

]

Anisotropic dihedrals� -

α = π2

Isotropicdihedrals[

1 00 −1

]

� � �� ) # ��. � �"� ��α

!�'�� ' � ��(�� !�� &'��

w, to the identity [1 0 0]t by means of the following transformations:

100

=

cosα sinα 0− sinα cosα 0

0 0 1

1 0 00 cos β sin β0 − sin β cos β

ejφ 0 00 ejδ 00 0 ejγ

w. (3.16)

The third matrix represents a set of scattering phase angles, whereas the first and thesecond ones are canonical forms of plane rotations. Physically only one of them, β,which ranges from −π to π, corresponds to a physical rotation of the sensor coordi-nates. Indeed, as long as we use the Pauli basis for the vectorization of the scatteringmatrix, β represents the physical orientation of the scatterer about the line-of-sight[Pap99]. Hence, by calculating β, one may obtain a direct estimate of the target ori-entation angles (a much simpler way than the polarimetric signature or the Stokesreflection matrix to determine the target orientation).

The parameter α is not related to the target orientation but represents an internaldegree of freedom of the scatterer. It is associated with the “type” of scattering mech-anism and can vary in the range [0, π/2]. α = 0 stands for isotropic surfaces, α = π/2for isotropic diplanes or helices. Low values of α represent all-anisotropic scatteringmechanisms with Shh different from Svv. The boundary between anisotropic surfacesand diplanes is represented by the case α = π/4, which describes an horizontal dipole.Finally, the information provided by α about the scatterer is independent of its ori-entation β, and hence unaffected also by eventual misplacements between radar andscatterer reference.

When multiple scattering mechanisms are present in the same coherency matrix(i. e., when more than one eigenvalue appearing in (2.128) is different from zero), thetarget is represented by [S] matrices that occur with the probabilities Pi just seen. In thiscase, a description of the overall scattering is possible by applying the parameterization(3.13) to each eigenvector of [T(3)] and then evaluating the mean of these parameters.In particular, the average α:

α = P1α1 + P2α2 + P3α3 (3.17)

results as the most useful quantity. Target behaviour may indeed be studied in a two-dimensionalH/α space. Here, since some limits exist to the variation of α as a functionof H , not all the space actually represents real scatterers (i. e., not all the values ofentropy and α have a physical meaning). The curves I and II of Figure 3.5 limit thefeasible space [CFLSS99]. In turn, boundaries can be drawn between sub-areas of this


�� &� �� H/α

� � !�'��

region referring to targets with well-defined characteristics (expressed by their valuesof α andH); the partition among various regions of theH/α space can actually be usedas a classification method as reported in [CP97].

As we have seen, the unitary matrices needed for the change-of-basis relations canbe derived only by means of the vectors of the bases involved, so that building a uni-tary matrix using three orthogonal vectors w yields 3 × 5 = 15 parameters and sixconstraints due to orthogonality: this means nine degrees of freedom (eight consider-ing also the special unitarity constraint, det[U(3)] = 1). Thus, bearing in mind thesecharacteristics of the change-of-basis matrices, it is possible to interpret the differencein number of independent parameters between the scattering and coherency (or co-variance) matrices. In order to fully describe these additional degrees of freedom, theyare no longer assigned to the “wave” representation, but to the projection matrices (theunitary matrices) of the scattering vectors onto the chosen basis of orthogonal vectorsw (i. e., to the chosen ensemble of scattering mechanisms).

3.4 Polarimetric SAR interferometry

This paragraph will introduce the theory concerning the use of polarimetric data forinterferometric applications. Here, only some basic concepts will be given and a moresophisticated treatment will follow later in this work.

Relatively few examples of interferometric applications of polarimetric SAR datahave been until recently reported in the literature. The main results in this field, towhich this section is dedicated, are those presented by Cloude’s research group [CP98],[Pap99], [PRC99a], [PRC99b]. At present, also the potential of interferometry withsingle-polarization systems is actively investigated (for example, the use of interfero-

3.4 - Polarimetric SAR interferometry 41

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� � ��

R

S

P

R+

B

1

2

S

∆

θR

� ��

� ��

�� &� �� ' � �� &. �� (� #�)$+-,/� . ! �&� '�� . ��

metric coherence images coupled with intensity images to improve thematic land clas-sification [Alb98], [BSA+99]).

The fundamental idea of radar interferometry consists of an accurate phase analysisof the radar signals backscattered by a target and acquired by two antennae placedin different positions, S1 and S2 , as represented in Figure 3.6. This situation can berealized both by using actually two antennae separated by a fixed distance and byusing only a single antenna mounted on a spaceborne or airborne device which getsdata twice of the same area while moving along two slightly different tracks. In thesecond case, besides the exact control of the phase of the transmitted MW beam, theknowledge of the position of the antenna is very important in order to determine thedistance, the interferometric baseline B , between the two “virtual” antennae.

The analysis performed on the signals follows the well-known rules also valid foroptical interferometry. In particular, when two complex images of the same area ac-quired from two slightly different points of view are processed together, the result isa fringe structure similar to those observed in optical experiments. This analogy is soexact that, for example, it is possible to define a direct parallelism between the Youngtwo slits experiment and the operation of a SAR interferometer:

• the partially coherent light source is replaced by the resolution cells on the groundwhich, when reached by the coherent radar signal, transmit its echo towards thetwo receiving antennae;

• the slits are substituted by the two antennae in the positions that they assumewhen they receive the pair of images used to synthesize the interferogram;

• the screen where the optic fringes appear is replaced by the whole electronic sys-tem of data acquisition and processing and by the one needed for the interfero-gram creation.


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As a result of the coupled imaging and interferogram synthesis, the interferomet-ric phase (i. e., the phase difference of the two signals coming from the same resolu-tion cell) and the interferometric coherence (which is a normalized expression of thisphase difference) may be calculated. The importance of these quantities lies in the factthat they are related to key parameters of the observed scene, for instance the verticalground elevation.

Let us consider at first a single-polarization processing. For each resolution cell oftwo images, I1 and I2 , of the same area, this involves a complex scalar signal of theform:

s = |s|ejδ. (3.18)

Thus, it is possible to define a Hermitian positive semidefinite coherency matrix as[CP98]:

[J] =

⟨[

s1

s2

]

[

s∗1 s∗2]

⟩

=

[

〈s1s∗1〉〈s1s

∗2〉

〈s2s∗1〉〈s2s

∗2〉

]

(3.19)

using the two involved signals.

From the elements of [J], the interferometric coherence and the expected interfero-


metric phase are derived in the following ways:

γ = |γ′| =|〈s1s

∗2〉|

√

〈s1s∗1〉〈s2s∗2〉=

|J12|√J11 J22

(3.20)

and

φ = arctan

(

Im〈s1s∗2〉

Re〈s1s∗2〉

)

. (3.21)

The term γ ′, also referred to as correlation coefficient, which represents the complexcross-correlation between the signals is introduced here as:

γ′ = γ exp(jφ) (3.22)

and the interferometric coherence γ is defined as its absolute value. With the definitionabove it holds as well that:

0 ≤ γ ≤ 1. (3.23)

To extend these concepts to vector signals, let us call k1 and k2 the scattering vectorsderived from the polarimetric data received from a resolution element of the imagedscene (for the sake of simplicity we drop the subscripts which refer to the dimensionsof the vector and to the chosen matrices basis). The expression of such vectors in thePauli matrix basis is given by Equation (2.114).

The analogue of [J] in this case can then be defined as:

[T(6)] =

⟨[

k1

k2

]

[

k†1 k

†2

]

⟩

=

[

[T11] [R12]

[R12]† [T22]

]

. (3.24)

[T(6)] is a block matrix, Hermitian and positive semidefinite, the elements of whichare the 3 × 3 matrices:

[T11] =⟨

k1k†1

⟩

, (3.25)

[T22] =⟨

k2k†2

⟩

(3.26)

and[R12] =

⟨

k1k†2

⟩

. (3.27)

According to their definition, [T11] and [T22] are the Hermitian coherency matricesof I1 and I2 (see Equation (2.116)), while [R12] is a complex matrix which takes intoaccount the interferometric phase relations of the different polarimetric channels be-tween both images. [R12] is not Hermitian since in general:

k1 6= k2 , (3.28)

hence:[R12] =

⟨

k1k†2

⟩

6=⟨

k2k†1

⟩

= [R12]†. (3.29)

The next step is to take two normalized complex vectors, w1 and w2 (see alsopage 38), to generate two complex scalars, µ1 and µ2 , as the projections of the scat-tering vectors k1 and k2 onto w1 and w2 respectively:

µ1 = w†1k1 , (3.30)


µ2 = w†2k2 . (3.31)

By means of these scalars it is possible to redefine [J] for polarimetric quantities byreplacing s1 and s2 in (3.19) by µ1 and µ2:

[J] =

⟨[

µ1

µ2

]

[

µ∗1 µ∗

2

]

⟩

=

=

⟨[

w†1k1

w†2k2

]

[

(w†1k1)

∗ (w†2k2)

∗]

⟩

=

=

[

w†1[T11]w1 w

†1[R12]w2

w†2[R12]

† w1 w†2[T22]w2

]

, (3.32)

where it must be noted that, for example:

(w†1k1)

∗ = (w†1k1)

†, (3.33)

since this is a scalar quantity.

Thus, the expressions of the interferometric coherence and of the expectation valueof the interferometric phase may again be derived from the elements of [J]:

γ =|〈µ1µ

∗2〉|

√

〈µ1µ∗1〉〈µ2µ

∗2〉

=|J12|√J11 J22

=

=|〈w†

1[R12]w2〉|√

〈w†1[T11]w1〉〈w†

2[T22]w2〉(3.34)

andφ = arg〈µ1µ

∗2〉 = arg〈w†

1[R12]w2〉 . (3.35)

About the w vectors one can note that using (2.114):

whh =

1/√

2

1/√

20

, (3.36)

wvv =

1/√

2

−1/√

20

(3.37)

and

whv =

001

, (3.38)

that is, according to (3.30) and (3.31), the hh linear polarization corresponds to a pro-jection of the given scattering vector onto the normalized vector whh and so on for theother polarizations.

Given these fundamental expressions, which provide us with a theoretical basis forstudying polarimetric interferometry, let us review what the main experimental resultsobtained to date have been.


In [CP98] and [Pap99] a method was presented to optimize the degree of correlationof an interferometric pair of images. It consisted of an eigenvalue analysis of the twoequations (with the same eigenvalues) derived by imposing a maximization constrainton (3.34). According to this technique, for each image of the pair, it is usually possibleto define three vectors w which are eigenvectors of these equations and which rep-resent different scattering mechanisms; the optimal coherence value is then obtainedusing the eigenvectors corresponding to the maximum eigenvalue. As the eigenvec-tors of a matrix are not uniquely defined, but do admit an arbitrary phase, one shouldadd an additional condition which fixes the phase difference between the two eigen-vectors w1 and w2 corresponding to the same mechanism; a sufficient one is to requirethat their phase difference be null, that is:

arg{w†1w2} = 0 . (3.39)

The physical meaning of the coherence optimization is the definition of scatteringmechanisms that would have the closest correspondence to a single point scatterer.

A very interesting result arises from the evaluation of the interferometric phase us-ing the unitary vectors w which maximize (3.34). As a major source of decorrelationresults to be that due to the height distribution of the scattering coefficient above thechosen reference plane [ZV92], [BH98], one sees that the coherence optimization partlyresolves just this height distribution. In fact, applying Equation (3.35) to the ith andjth mechanism and calculating its difference, i. e.:

∆φij = arg{µ1iµ∗

2i} − arg{µ1j

µ∗2j}, (3.40)

it is in some cases possible to measure the height difference associated to ∆φij . Whenapplied to the estimation of forests height, this approach shows limits that depend onthe properties of the scattering components. Indeed, while propagation through a ran-dom volume (that of the branches and layers of leaves where, usually, no preferentialdirection is present) is unaffected by changes of polarization, the ground scattering ispolarization dependent; but as no polarization presents only one of the two contribu-tions, it is not possible to completely separate them and estimate their difference inheight through their interferometric phase difference [PRC99a], [PRC99b].

We have already said that, at present, few reports on applications of polarimetricSAR interferometry have been published; hence, this section has provided only a short,but fairly up-to-date description of the state of the art in this study area. Further origi-nal examples of such applications will be presented in Chapter 5.

The research related to scalar interferometry is, instead, more advanced. Thoughsome of its features are similar to vector interferometry (for instance, phase unwrappingmethods) we will not deal with them in this work in order to focus our attention onpolarimetry. Nevertheless, starting from scalar technology achievements, several newtopics can be suggested for future vector applications such as differential polarimetricinterferometry, as proposed in [Pap99], or the use of polarimetric coherence images forpattern recognition and classification.

4 Analysis of polarimetric parameters:image classification

4.1 Introduction

As already seen, an electromagnetic wave may be described in different ways and eachof them has a particular representation of its scattering interactions. Due to the vecto-rial nature of EM waves, the interactions are expressed in terms of matrices and againsome matrices connect the various descriptions. We have also seen that some rules ex-ist which impose constraints on these representations and on the transformation matri-ces, so that the number of the independent parameters, i. e., the carriers of physicallyrelevant information, remains constant. The next question becomes: are the variousrepresentations absolutely equivalent? Why choose a particular representation? Arethere parameters which are intrinsically more significant than others?

The answers to these questions should take into account also other considerations.In fact, the choice between two or more representations depends on the target to bedetected and, in this sense, no absolute best descriptor is given. In the context of classi-fication applications, the decision whether to use a linearly or an elliptically polarizedbase representation may depend on the fact that one is more interested in detecting,for example, buildings rather than trees. Hence, each problem has to be solved indi-vidually.

In the following, we will analytically study the various polarimetric parameters, try-ing to make this review as comprehensive as possible. In order to compare them andhave an objective estimate of how well they may describe a certain target, some clas-sification tests will be performed. In other words, an attempt to measure the amountof information provided by the different polarimetric parameters is made. It shouldbe stressed that the main topic of our research will remain the polarimetric parame-ters, rather than the adopted classification techniques (of which only the indispensablenotions will be given for sake of completeness) and their results; indeed, their exactknowledge will prove itself useful to better understand the scattering phenomena inconnection with the characteristics of the wave and of the representation used.

The rest of this chapter is organized as follows: in the next section, the basics on clas-sification theory and on the classification algorithms used in this work will be given;then the parameters under study will be briefly reviewed and the experimental setupdescribed; finally the results of the tests will be reported and discussed for each polari-metric parameter considered.

47

48 Chapter 4 - Analysis of polarimetric parameters: image classification

4.2 Classification theory and accuracy assessment

Let us introduce here those basic concepts of image classification which are necessaryto understand the tests performed later on in this chapter. Among others, the classifi-cation algorithms used will be described and some methods for classification accuracyassessment explained.

4.2.1 Classification theory basics

In general, the classification of an image is a process implying the identification of thedifferent spectral classes1 present in it and their connection to some specific groundcover types [Ric94]. Usually, this is accomplished using general mathematical patternrecognition and pattern classification techniques, though more specific methods havebeen developed especially suited for various sensors (as in the case of one based on theH/α parameters for polarimetric SARs [CP97]). The patterns are the pixels themselvesthat contain the values corresponding to the measured parameters; it is a commonpractice to represent them through column vectors like:

x =

x1

x2...xn

, (4.1)

where x1, x2, · · · , xn are the spectral values of the pixel vector x in bands 1 to n, respec-tively. Therefore, classification involves labelling the pixels as belonging to particularspectral (and thus information) classes using the spectral characteristics of the avail-able data.

During classification, each pixel of an image is no longer identified by its position inthe chosen spatial reference system but by means of the vector belonging to the featurespace relative to the spectral values. One then establishes a set of rules to label thepixels according to their particular properties in this space. To derive these rules, it isalso necessary to define boundaries among the different classes in the n-dimensionalfeature space.

Two main types of classification procedures may be identified: the supervised andthe unsupervised classification methods. Each type finds application in the analysis ofremote sensing image data. These may be used as alternative approaches but are alsooften combined into hybrid methodologies.

Unsupervised classification is a means by which pixels are assigned to spectralclasses without foreknowledge of the existence and properties of classes. Most often itis performed using clustering methods. These procedures can be used to determine atfirst the number and location of the spectral classes into which the data fall and thenthe spectral class of each pixel. The user finally identifies those classes a posteriori, byassociating a sample of pixels in each class with available data, for instance maps andinformation from ground truth.

1“Spectral classes” are defined by similar image pixel values, which may be related to the variousground coverages in a scene.

4.2 - Classification theory and accuracy assessment 49

Clustering procedures are generally computationally expensive, they are neverthe-less central to the analysis of remote sensing imagery. An interesting aspect of thesemethods is that whilst the ground cover classes, for a certain application, are usuallyknown, the corresponding spectral classes are not. Unsupervised classification is there-fore useful for determining spectral class composition of the data prior to any furtherdetailed analysis with supervised methods. Examples of proposed algorithms for un-supervised classification are the K-means and the isodata [TG74]. Other approaches,not based on clustering, use inherent characteristics of the data used; for polarimet-ric SAR data, are worth mentioning the already cited methods by van Zyl [vZ89] andCloude and Pottier [CP97].

Supervised classification is the procedure most often used for quantitative analy-sis of remote sensing image data. It is usually based on an important assumption,namely that each spectral class can be described by a probability distribution in themultispectral space; this will be a multivariable distribution with as many variables asthe dimensions of the space. Such a distribution describes the chance of finding a pixelbelonging to that class at any given location in the multispectral space. Almost all thealgorithms used for this purpose consist of the same series of basic steps:

1. decide the set of ground coverages into which the image is to be segmented.The examples in this work are the classes: “water”, “houses”, “roads”, “trees”,“grass”, “field 1” and “field 2”.

2. A set of prototype pixels, the training data, is then selected for each class of thedesired set. This choice may be done using information from ground surveys,aerial photographs, topographic maps or any other source of reference data.

3. The following step consists of describing each class by means of some particularparameters, mainly statistical ones such as the first and second order moments,depending on the chosen classifier. These parameters characterize the adoptedprobability model or directly define the partition of the multispectral space. Theset of parameters for a given class is sometimes called the signature of that class.

4. Using the trained classifier labels, every pixel of the image is assigned to one ofthe desired ground cover types (information classes). In practice, the completeimage is automatically classified after the user has directly identified just a por-tion of it in the second step when defining the training data.

5. Produce tabular summaries or thematic maps summarizing the classification re-sults.

A variety of algorithms that perform supervised classification are available: paral-lelepiped, maximum likelihood, minimum distance, Mahalanobis classifier, etc. Here,we will briefly describe only the first three methods as they are those used in this work.

4.2.2 Parallelepiped classification

The parallelepiped (Parall.) classifier is a very simple supervised classification technique;in principle, the classifier is trained by inspecting histograms of the individual spectral


components of the selected training data. The upper and the lower significant boundsof the histograms are identified as the edges of a multidimensional parallelepiped foreach class. If, on classification, pixels are found to lie in such a parallelepiped they arelabelled as belonging to that class.

Relevant problems of this criteria are that pixels in regions between parallelepipedscannot be assigned to any class and pixels in overlapping areas cannot be separated:in both cases, these pixels are not classified. No statistical costraints in the definition ofthe class membership are taken into account here.

4.2.3 Maximum likelihood classification

Maximum likelihood (Max. Lik.) classification is based on the estimation of the statisticaldistributions of the spectral components [Ric94]. This allows to evaluate the chance offinding a pixel belonging to a certain class at any given location in the multispectralspace.

The working hypothesis usually adopted is that the probability distribution for eachclass is a multidimensional normal, or Gaussian, distribution. Hence, the probabilityof finding a pixel belonging to a given class ωi in a position x of the multispectral spaceis described by a function like:

p(x|ωi) =1

(2π)n/2(det [Σi])1/2exp

{

−1

2(x − mi)

t [Σi]−1(x − mi)

}

, (4.2)

where i = 1, 2, . . . , l is the index identifying the l ground cover classes, mi the meanvector of the data in class ωi and [Σi] the covariance matrix for that class. For each i, the[Σ] matrix is defined as:

[Σ] = 〈(x − m) (x − m)t〉 =1

r − 1

r∑

j =1

(xj − m) (xj − m)t, (4.3)

being r the total number of pixel of the class. The multidimensional normal distribu-tion is completely specified by its mean vector and its covariance matrix.

The covariance matrix introduced here is different from the one associated to polari-metric data. It allows, for instance, to see if a correlation exists between the responsesof a pair of spectral bands: if they are correlated, the corresponding off-diagonal ele-ments of the covariance matrix are large with respect to the diagonal terms (whereaslittle correlation yields off-diagonal terms close to zero). Let us now see how thesehypotheses and expression are actually used by the classifier.

In trying to determine the class or category to which a pixel characterized by a vectorx belongs, it is strictly the conditional probabilities

p(ωi|x), i = 1, 2, . . . , l (4.4)

that are of interest. As already said, the vector x describes the pixel as a point in themultispectral space with coordinates defined by the values of the parameters. Theprobability p(ωi|x) gives the likelihood that the correct class is ωi for the pixel at po-sition x. Hence, the classification is performed assigning a pixel vector x to class ωi


when the condition:p(ωi|x) > p(ωj|x) ∀ j 6= i (4.5)

is verified. The problem is then to estimate the probabilities p(ωi|x) that are unknown.

If sufficient training data are available for each class (each ground cover type de-fined by the user), the probability distribution, which describes the chance of finding apixel from ωi at the position x, can be calculated. This new probability distribution isindicated with p(x|ωi). The desired p(ωi|x) and the estimated p(x|ωi) are related by theBayes theorem [Pap65]:

p(ωi|x) =p(x|ωi)p(ωi)

p(x), (4.6)

where p(ωi) is the probability that class ωi occurs in the image and p(x) is the probabil-ity of finding a pixel from any class at the location x. The p(ωi) are called a priori or priorprobabilities, since they express the probabilities with which the class membership ofa pixel could be predicted before classification. By adopting the same interpretationprinciple, the p(ωi|x) are posterior probabilities.

Using (4.6), the classification rule of (4.5) may be written as:

x ∈ ωi if p(x|ωi)p(ωi) > p(x|ωj)p(ωj) ∀ j 6= i. (4.7)

Mathematical convenience results if discriminant functions like:

gi(x) = ln{p(x|ωi)p(ωi)} = ln{p(x|ωi)} + ln{p(ωi)} (4.8)

are used, so that (4.7) is restated as:

x ∈ ωi if gi(x) > gj(x) ∀ j 6= i. (4.9)

At this stage it is assumed that the probability distributions for each class are of theform of the multivariate normal model, see (4.2). This is an assumption, rather than ademonstrable property for natural spectral or information classes; however it is cho-sen because it leads to mathematical simplifications and moreover it is a distributionfor which properties of the multivariate form are well-known. The final form for thediscriminant function, based upon the assumption of normal statistics, is then:

gi(x) = ln p(ωi) −1

2ln(det[Σi]) −

1

2(x − mi)

t[Σi]−1(x − mi). (4.10)

The boundaries among different classes, because of the second order equation thatcharacterizes the discriminant function, are quadratic curves such as parabolas, circlesand ellipses. As will be shown later, the boundaries in the minimum distance approachare expressed by linear equations, hence, the higher order decision rules make themaximum likelihood classification more efficient for partitioning multispectral spacethan the minimum distance one.

It is also possible to introduce a threshold in the case that insufficient training dataare available to estimate the parameters of the class distributions; the threshold of eachclass is calculated from the knowledge of its a priori probability and its covariancematrix.


4.2.4 Minimum distance classification

The effectiveness of maximum likelihood classification is related to the accuracy in theestimation of the mean vector m and the covariance matrix [Σ] for each spectral class[Ric94]. This, in turn, depends upon having a sufficient number of training pixels foreach of those classes. If this condition is not satisfied, an inaccurate estimate of [Σ]results, leading to poor classification results. When the number of training samplesper classes is limited, it is then worth trying to use a classification approach based onlyon the use of m, whose evaluation, even for small quantities of samples, can be moreaccurate than that of the covariance matrix elements. A classifier which needs only themean vectors of the various classes to be estimated is the minimum distance (Min. Dist.),which should be more precisely indicated as minimum distance to class means classifier.Indeed, this algorithm operates by placing a pixel in the class of the nearest mean inthe multispectral measurement space.

An advantage of this method with respect to the maximum likelihood classificationis its computational speed. The main disadvantage stems from the fact that it does notuse the covariance matrix and it therefore models only symmetric classes in the mul-tispectral space; hence, classes with a well-defined data spread in a particular spectraldirection cannot be represented correctly.

The discriminant function for the minimum distance classifier is derived from theexpression of the squared Euclidean distance between the position of the generic pixelx to be classified and the mean value mi of i-th class:

d(x,mi)2 = (x − mi)

t(x − mi). (4.11)

Expanding the product gives:

d(x,mi)2 = xt x − 2mt

i x + mti mi . (4.12)

Classification is performed on the basis of:

x ∈ ωi if d(x,mi)2 < d(x,mj)

2 ∀ j 6= i . (4.13)

As xt x is common to all d(x,mi)2, it can be neglected. By reversing the signs of the

expressions above, the assignment to a class is again expressed by the condition (4.9)with:

gi(x) = 2mti x − mt

i mi . (4.14)

As a consequence of the linearity of the discriminant functions, the decision surfacesfor this classifier are planes in the n-dimensional spectral space. The minimum distanceclassification is proven to be of value when the number of training samples is limitedand, in such a case, can lead to better results than the maximum likelihood procedure.

Finally, the possibility of adopting a definition for the distance other than the Eu-clidean is also worth mentioning.

4.2.5 Classification accuracy assessment

The final step in any classification procedure is an accuracy assessment. This shouldnot be meant as an afterthought but as an integral part of any classification project,which explains and completes the results of the classifier used.


The most common way to represent the classification accuracy of remotely senseddata is in the form of an error matrix (also called confusion matrix). This is a square arrayof numbers set out in rows and columns which express the number of sample units(in this work, pixels) assigned to a particular category relative to the actual categoryas verified on the ground. Usually, the columns represent the reference data whilethe rows indicate the classification generated from the acquired data. Let us refer,for example, to Table 4.1 that reports the results relative to one of the tests discussedlater. Considering the class “water”, one can evaluate how its sample pixels havebeen classified by studying the corresponding column of the table. Only 39.73% of thesample pixels have been correctly recognized as belonging to that class; 42.33% havebeen assigned to the class “roads” and no pixels at all have been classified as “houses”or “trees”. The row of the table gives information on the misclassification of the othersample pixels, so that one sees that 15.13%, 18.99% and 8.19% of the sample pixelsof the classes “roads”, “grass” and and “field 1”, respectively, have been incorrectlyclassified as “water”.

The confusion matrix is a very effective way to represent accuracy as much as theaccuracies of each category are plainly described along with both the error of inclusion(commission errors) and error of exclusion (omission errors) present in the classifica-tion [Con91], [Ste97].

The error matrix can then be used as a starting point for deriving a series of descrip-tive and analytical statistical techniques. Perhaps the simplest descriptive statistic isthe overall accuracy (Ov. Acc.), which is computed by dividing the total of correctly as-signed pixel (i. e., the sum of the major diagonal), by the total number of pixels inthe error matrix. In addition, accuracies of individual categories can be computed ina similar manner. Traditionally, the total number of “correct pixels” in a category isdivided by the total number of pixels of that category as derived from the referencedata (i. e., the column total). This accuracy measurement indicates the probability of areference pixel being correctly classified and is really a measure of the omission error:it is often called producer’s accuracy (Pr. Acc.) because the producer of the classificationis interested in how well a certain area is classified. On the other hand, if the total num-

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Class water houses roads trees grass field 1 field 2 Totalwater 39.73 0.02 15.13 0.04 18.99 8.19 14.04 13.27

houses 0.00 59.26 0.00 4.33 0.00 0.00 0.10 9.93roads 42.33 0.02 76.21 0.01 19.74 7.95 14.53 21.72trees 0.00 35.42 0.00 76.42 0.23 2.91 0.69 17.63grass 14.79 0.06 4.51 0.18 23.00 12.70 20.03 10.19field 1 1.85 5.04 1.80 18.35 23.52 50.44 27.49 19.00field 2 1.29 0.19 2.35 0.66 14.52 17.82 23.12 8.26Total 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00


ber of “correct pixels” in a category is divided by the total number of pixels classifiedin that category, then this result gives a measure of the commissions. This measure,called user’s accuracy (Us. Acc.), is indicative of the probability that a pixel on the imageactually represents that category on the ground.

It is also common practice to normalize the error matrices by an iterative propor-tional fitting procedure that forces each row and column in the matrix to sum to one.In this way, differences in sample sizes used to generate the matrices are eliminatedand, therefore, individual cell values within the matrix are comparable. In addition,the resulting normalized matrix is more indicative of the off-diagonal cell values (i. e.,the errors of omission and commission).

Another measure of the accuracy of a classification procedure is the Kappa coefficientdefined as:

K =

Nr

∑

i=1

xii −r

∑

i=1

(xi+ · x+i)

N2 −r

∑

i=1

(xi+ · x+i), (4.15)

where r is the number of rows in the matrix, xii is the number of observations in rowi and column i, xi+ and x+i are the marginal totals of row i and column i, respectively,and N is the total number of observations. For 0.4 < K < 0.75 the accuracy of theclassification is good and it is excellent for K > 0.75 [Con91].

In general, it is not possible to give clear-cut rules indicating when each measureshould be used. Each accuracy measure incorporates different information about theerror matrix and therefore must be used as a different computation attempting to ex-plain the errors performed in the classification.

4.3 Overview on polarimetric parameters

Now that the theoretical bases to our tests and to their interpretation have been pro-vided, it is possible to explain in detail the adopted analysis strategy and commentupon the results.

As a first step, it is helpful to summarize the parameters that can be extracted froma fully polarimetric SAR data set, i. e., those different ways in which these data can bevisualized and analyzed:

Backscattered intensity. This is the observable directly detected by the sensor andgives the most immediate picture of the observed scene. The use of differentpolarization bases can be meaningful as far as intensities are concerned.

Ratios of the [S] matrix elements. One may use this approach in order to enhance andstudy the relative behaviour of the polarization channels. In particular, in thiswork, we will calculate for SAR data some quantities usually derived for weatherradar data, like the linear depolarization ratios (LDR), i. e., the ratios between cross-polar channels and co-polar ones, and the differential reflectivity (ZDR), that is, theratio between two co-polar channels (an example of a ZDR image is shown inFigure 4.5).

4.4 - Experimental approach 55

Characteristic polarizations. The co- and cross-polar nulls of the scattering matrixmay be calculated to enhance image quality and to retrieve orthogonality prop-erties of the polarizations.

Parameters of the coherent target decomposition theorems. In general, the applica-tion of such theorems leads to pictures that express the intensity of various scat-tering mechanisms. Higher or lower intensity values of one component are con-nected to the predominance of a mechanism with respect to the others.

Entropy/α parameters. This approach involves the generation of the coherency ma-trix and the calculation of the eigenvalues and other related quantities. Theirinterpretation follows the principles proposed by Cloude and Pottier [CP97].

One should note that, although each parameter provides relevant information aboutthe scatterers, none seems to be sufficient to fully interpret the response of a scene onits own, so that their joint use is a common practice. In the recent research literature theuse of Entropy/α method for classification purposes has been reported and carefullyinvestigated [CP97], [Hel00]. The application of coherent methods for achieving sim-ilar purposes, however, appears to have remained neglected and basically untested.This is the reason why, in this contribution, we wish to address this issue, in an at-tempt to complete the knowledge of all the considered polarimetric parameters.

4.4 Experimental approach

A series of classification tests has been carried out for each one of the different ap-proaches listed in Paragraph 4.3 [AC00], [PAC+02], [ACP02b]. Since the main interestof this research is not the theory of pattern recognition and image classification itself,we have considered these techniques simply as a tool to compare different polarimetricparameters or, rather, to have a first numerical estimation of their “potential”. Hence,we have chosen classification algorithms widely used in the field of remote sensing,though not specifically intended for SAR data, namely the parallelepiped, the min-imum distance and the maximum likelihood. As previously seen, these algorithmsassign the pixels of the images to selected ground cover types according to some statis-tical quantities (mean, variance, maxima and minima) extracted from prototype pixelsbelonging to these classes. For the Parall. and Min. Dist. classifiers no further assump-tions are made on the statistical properties of the data or of the pixels belonging to theclasses in particular. Conversely, the Max. Lik. algorithm is based on the hypothesis,reasonable in most of the cases for remotely sensed data, that the probability distribu-tions for each class are in the form of multivariate normal models. Since this conditionis hardly met for SAR polarimetric data [FMYS97], the use of this algorithm is contro-versial and the results presented here must be carefully interpreted.

An original aspect of this work is the study of the dependence of the classificationresults on the size of averaging windows of pixels, i. e., all the tests have been per-formed using images where the values of the pixels have been averaged consideringsquare windows of increasing size, namely: 3×3, 5×5, 10×10 and 15×15 pixels. Suchan analysis will permit to prove if the chosen polarimetric parameters provide a de-scription only of “point-like” physical properties of the targets or if they also contain


“extended”, local information. Indeed, in the first case we expect good accuracy onlyfor single-pixel or small averaging windows based classification, whereas the presenceand recognition of extended parameters characteristics would lead to enhanced classi-fication accuracy also for larger averaging windows.

Finally, the comparison among different polarimetric parameters will contribute tothe search of targets and/or ground cover classes that are better recognized and un-ambiguously characterized by one parameter rather than another.

Regarding the outlined classification approach, it may be objectionable that theadopted techniques may not be adequate to the data (as we admitted for the Max.Lik. algorithm) and that more sophisticated approaches could better account for thestatistical properties of targets. Similar doubts could be raised also by the fact that lit-tle or no attention was paid to noise removal or speckle reduction. However, there is aheuristic component in any polarimetric approach developed to date and, at this point,results seem to justify assumptions. Thus, by means of our tests, it is still possible to ac-complish our tasks: to study the influence of the averaging windows dimensions andto broadly establish what kind of features are better detected by a given polarimetricparameter.

4.5 Data sets and test areas

The experimental data used for our tests are the following:

• Single-look complex (SLC) data sets of the area of Oberpfaffenhofen, Germany,acquired by the E-SAR airborne sensor operated by DLR during a measurementcampaign in October ‘99. The data consist of L-band scattering matrices mea-sured in the hv-basis.

The imaged area is situated approximately 25 km South West of the city ofMunich and includes several interesting features: the DLR centre, the formerFairchild Dornier aeroplane factory and the airfield shared by the two firms (seeFigure 4.2).

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RF-band LCentre frequency 1.3 GHz

Wavelength 23 cmBandwidth 100 MHz

Mode Quad Pol (hh, vh, vv, hv)PRF 400 Hz (per channel)

Transmit peak power 360 WAntenna gain 17 dB

Range resolution 1.5 mAzimuth resolution 0.89 m

4.5 - Data sets and test areas 57

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Not far from them is situated the town of Gilching. Other important man-madefeatures are the motorway and the railway line stretching across the image. Thevegetation patches consist of coniferous and mixed forests, meadows and crops.

• Similar data of the same sensor acquired over the urban area of Munich. Thescene contains mainly buildings, but green areas are also present.

Aerial photographs were also used as ground truth and complementary sources ofinformation.

About the ground cover classes, we will concentrate on those types that are usuallyinvestigated in the literature:

• Urban areas and man-made artifacts (even when isolated);

• forests (trying to distinguish among dense, coniferous and sparse, deciduousones);

• sparse vegetation;

• agricultural crops;

• water.

Whenever possible, some other distinctions have been made inside the types de-scribed above in order to refine this broad grouping (for example, to try to recognize


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different crop types or woods). This was done, in particular, when using the [S] matrixelements ratios.

The optical images served as ground truth data, since no more precise informationtaken on the ground was available. The classes mentioned (forests, water, grass, etc.)have been identified by direct interpretation of these aerial photographs. Regardingthe class indicated later on as “houses”, this definition refers to quite homogeneousareas characterized by family houses surrounded by gardens (often including trees).

4.6 - Backscattered wave amplitude 59

4.6 Backscattered wave amplitude

A common analysis procedure was adopted for all the polarimetric parameters. Atfirst, seven ground cover classes (a more refined set with respect to the coarse one pre-viously outlined) were defined: “water”, “houses”, “roads”, “trees”, “grass”, “field 1”and “field 2”. Then, the corresponding prototype pixels from each class were selectedand passed as training samples to each classification algorithm. Finally all of the datawere classified. These last two steps were repeated five times, on a single-pixel basisand also considering the 3×3, 5×5, 10×10 and 15×15-pixel averaging windows. Tablessummarizing the most meaningful results of all the tests are reported in Appendix C.

In this section we will start by analyzing the backscattered wave amplitude, i. e., thedata directly detected by the sensor and which provide the most immediate picturesof the observed scene [PAC+02]. The series of tests was performed giving the threeamplitude images in the hv-linear basis, |Shh|, |Svv| and |Shv|, as input to the classifiers.Only the data of the Oberpfaffenhofen test site were taken into account.

Figure 4.3 shows the classification accuracy assessments in terms of overall accu-racy and Kappa coefficient. It is clear that increasing the dimensions of the averagingwindow the classification accuracy improves; nevertheless, almost all the methods, in-dependent of window dimensions, do not reach satisfactory accuracy degrees. Indeed,they all remain below or near the threshold of K > 0.4 which indicates a good classi-fication accuracy and only the Max. Lik. classifier, for window dimensions larger than5×5 pixel, clearly exceeds this threshold.

Let us limit ourselves to the case corresponding to the best results: the 15× 15-pixel window. The classes giving rise to the main recognition problems are: “water”,“roads” and “grass” (see Appendix C.1). These are the ones corresponding to groundcover types and samples giving the lowest backscattered intensity. The “grass” train-ing pixels correspond to the flat area near the runway that, although rougher than theother two, still tend to reflect most of the incoming MWs away from the backscatteringdirection. Basically, the bad classification results for these three classes are due to theirlow signal-to-noise ratio (SNR). In particular, the Parall. classifier, the algorithm based

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on the simplest feature space partition, fails to recognize all the samples belonging tothem.

The best results in terms of both the producer’s accuracy and the user’s accuracyare obtained for the class “trees”: 85.42% and 64.13%, respectively, for the Parall. and76.42% and 65.08% for the Min. Dist. classifier. This means that most of the pixelsbelonging to this class have been correctly classified and that few samples from othertraining areas have been erroneously labelled as “trees”. Hence, it seems that the [S]matrix elements amplitudes are sufficient to characterize the targets associated withthat ground cover type. Conversely, none of the other classes yields such good results;

4.7 - Ratios of the scattering matrix elements 61

for the “houses”, whereas the Us. Acc. is quite high for all the classifiers, the Pr. Acc.remains low.

As shown in the accuracy assessment graphs, for smaller averaging windows, thevalues of Kappa and Ov. Acc. are such that the classification results cannot be consid-ered valuable.

4.7 Ratios of the scattering matrix elements

Let us now consider the backscattered intensity data from a different point of view andconcentrate on the relative behaviour of the polarization channels [AC00]. In order todo this, some simple mathematical ratios will be used, namely the linear depolarizationratios:

LDR(1) = 10 log|Shv|2|Svv|2

= 10 logσhv

σvv

(4.16)

and

LDR(2) = 10 log|Shv|2|Shh|2

= 10 logσhv

σhh

(4.17)

and the differential reflectivity:

ZDR = 10 log|Shh|2|Svv|2

= 10 logσhh

σvv

. (4.18)

As already mentioned, these polarimetric observables have been extensively usedin weather radar applications: for classifying hydrometeors [SCM88] and for estimat-ing meteorological quantities like the drop size distribution and rain-rates [SBAK79]via solving inversion problems. Particularly relevant are propagation studies, includ-ing attenuation studies and studies on the depolarization behaviour of scatterers, likeraindrops or ice crystals, along a propagation path [Hol84]. However, to our knowl-edge, the application of the ratios of the [S] matrix elements to the remote sensing ofthe Earth surface has been limited [ROvZJ93], [DvZE95].

Some other ratios could also be considered, such as those relating the [S] matrixelements expressed in different bases, for example, the RL-circular one as reported in[ROvZJ93].

The classification results obtained using different averaging windows do not leadto meaningful accuracy values for any of the classification algorithms. This may bedue to the very limited variance of the ratio values among the classes. To verify this,we decided to study the polarization ratios more thoroughly, defining an extended setof classes for the Oberpfaffenhofen test site and using also other data relative to theMunich urban area.

The training areas selected in the Oberpfaffenhofen scene were labelled as follows:“water”, “houses”, “bare asphalt”, “trees”, “grass”, “field 1”, “field 2”, “field 3”,“field 4” and “field 5”. The “bare asphalt” is represented by the airfield, whereas thenumber of fields, which have different optical behaviour, have been extended to five.One can broadly distinguish harvested soil from not harvested but, unfortunately, thelack of direct information on the ground, like crop height or humidity, does not allow


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to fully explain the different backscattering signals. Hence, it is only possible to notethe variations of the values in the images and register them without providing but ahypothetical interpretation.

In the Munich test site, the total number of training areas is lower than in the Oberp-faffenhofen one. Again, areas representative of the classes “water”, “houses”, “trees”and “grass” could be found. The class “buildings” has been introduced to define somebigger buildings, blockhouses and big sheds, which could in no way be associated withthe class “houses” and whose response, especially in the hh-channel, is particularlyhigh. We reiterate here that the class “houses” was associated to quite homogeneousareas characterized by family houses surrounded by gardens.

Figure 4.6 (a) reports the mean polarization ratios relative to a 5×5-pixel averagingwindow, whereas the values in Figure 4.6 (b) refer to single-pixel estimates. In bothcases, it is evident that the mean values of the three ratios do not vary significantlyamong the various classes. Considering also the standard deviations, to have a rangeof variation for each class, one can see that these ranges always overlap. This confirmsour hypothesis and explains why none of the ratios suffices to distinguish a class fromthe others (at least among the ones we have defined here). As a consequence, classi-fication based on these images only is unlikely to provide valuable results. Referring,for instance, to the quantities averaged on a 5×5-pixel window, the test with the Min.Dist. classifier gives for the Oberpfaffenhofen site an Ov. Acc. around 28 % and a valuefor the Kappa coefficient of 0.242, and for Munich a slightly better result of Ov. Acc. =

4.7 - Ratios of the scattering matrix elements 63

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57.4 % and K = 0.476 (results far from, or just at the limit of, the range 0.4 < K < 0.75which indicates a good accuracy).

Almost everywhere the hh-channel response is slightly higher than the vv-one.Hence, it seems that a real polarization isotropy of the backscattered signal, of whichthe ZDR is a function [SCM88], is rarely present.

According to the theory [CM91], [BW85], [UMF82], the predominance of one termover the other is connected to the type of scattering on the illuminated surface. Moreprecisely, |Shh| > |Svv| when the incident beam is scattered according to the Fresnelmodel, valid for surfaces almost flat, whereas the case of |Shh| < |Svv| is verified forscattering from rough surfaces described by the Bragg model. The hypothesis of Fres-nel scattering applies, for example, to the “buildings” (ZDR high and LDR(2) low).Also the class “bare asphalt” seems to follow this trend, although with less pronouncedevidence. Conversely, the only observed case of significantly negative ZDR average(and of LDR(1) values lower than the LDR(2) ones) refers to a field where we maysuppose a recent ploughing and, for this reason, an enhanced roughness.

The class “trees”, from which we expected one of the highest average values for theLDR ratios (assuming the presence of the best conditions to have depolarization effectsdue to multiple scattering phenomena) shows clearly this tendency in the Oberpfaffen-hofen test site and less evidently in the Munich test site. The reason for this differencecan be the type of trees present in the two scenes: mainly coniferous (at least in theselected sample) in the Oberpfaffenhofen scene, deciduous in the Munich sample. Onthe acquisition date, beginning of October, the trees in Munich whould have alreadylost their leaves and one can expect from them a less probable evidence of multiplebounce (and hence depolarization) phenomena.


4.8 Characteristic polarizations

The co- and cross-polar nulls were introduced in Paragraph 2.3 where we saw that bytheir means it was possible to express the scattering matrix in diagonalized forms, andthereby to better deal with representations of the interactions where the backscatteredpower is concentrated only in some of the components of [S].

Referring more in particular to the results of the zeroing constraints, we obtainedtwo solutions, i. e., two different matrices, with cross-polar null elements (see Equa-tions (2.88) and (2.89)) and four with co-polar null elements ((2.94), (2.95), (2.96) and(2.97)). These last four matrices differ only in phase terms (see (2.98) and (2.99)). Thus,we have selected only some elements of the derived matrices to perform our tests, na-mely, the two complex terms appearing in (2.88) which were indicated as p1 and q1 andthe two of (2.94) named x1 and a1. The amplitudes of the four chosen elements havebeen provided as inputs to the classifiers with the usual repeated tests on differentaveraging window dimensions.

When using the co- and cross-polar nulls one gets, in general, better classification re-sults than with the original amplitudes of the [S] matrix elements [PAC+02], [ACP02b].This is more evident for the larger averaging windows, as can be seen in Figure 4.7. Forwindows of 5 × 5 pixel or larger, the value of Kappa of the Min. Dist. and Max. Lik.classifiers is almost equal to or greater than 0.4, so that their results can be consideredacceptable (as did not happen for the tests on the original data).

We can comment here in detail on the behaviour of the various classes for the testswith the 15×15-pixel window (see Appendix C.2). The best classification results areobtained for the classes “houses”, “trees” and “field 1”. Indeed, for these classes, thepercentages of both the user’s accuracy and producer’s accuracy remain quite highfor all the classification algorithms (the lowest value is of 63.24% for the Prod. Acc.of the “houses” with the Min. Dist. classifier). Most of the pixels belonging to theseclasses have been correctly classified and few samples from other training areas havebeen erroneously assigned to the classes “houses”, “trees” or “field 1”. Hence, it seemsthat the co- and cross-polar nulls can be used to describe the targets associated with

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4.8 - Characteristic polarizations 65

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(c)

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those ground cover types. On the contrary, more controversial is the classification ofthe “grass” and “field 2” classes, for whom the accuracy estimates remain low even inthe considered best case of 15×15-pixel averaging window. The class “roads”, finally,seems to represent a problem only for the Parall. classifier that gives a very low value,3.65%, of producer’s accuracy.

For windows smaller than 15×15 pixels or for single-pixel based tests, the accuracyestimates drop rapidly and only the class “trees” shows user’s and producer’s accuracyvalues higher than 50%.


4.9 Parameters of the coherent target decomposition the-orems

In Paragraph 2.5, we presented a review of TD theorems and stressed how these meth-ods “intrinsically” represent a classification tool since they recognize and weight thecontributions of different targets in a scene. For example, a colour composite image ofthe Krogager decomposition coefficients is already a classification image: the identi-fied features are the three scattering mechanisms contemplated in the model and theirweights are a way to assign the image pixels to each of them.

These methods have been used here as a preliminary step to the supervised classifi-cations also carried out with the other polarimetric parameters. In particular, we stud-ied [ACP02b] the three main coherent decomposition theorems: the one by Krogager,the Pauli one and that of Cameron. Since they are all coherent methods, they provideinformation on a pixel-level basis and refer to point-like scatterers. Notwithstandingtheir original point-like nature, it is again worth investigating how the significance ofthese decomposition theorems stretches over extended areas. Hence, for all of the TDmethods, we repeated our tests with the three classification algorithms: firstly on thedata directly derived through a given decomposition, and secondly after averaging therelevant parameters. Results are reported in Appendix C.3.

The first TD theorem under consideration is the SDH decomposition [Kro93], [KC95].As seen on page 27, this approach yields the decomposition of the scattering matrixinto three components, as if the scattering were due to a sphere, a diplane and a right-or left-wound helix. The relative weight of each contribution is determined by the ki

coefficients appearing in Equation (2.121); they are real quantities and provide threenew pictures of the imaged scene. The classification tests have been performed onlyon these values provided as a three-layer input to the classifiers.

The overall accuracy estimates present values for the pixel based tests similar tothose previously seen with the other polarimetric parameters. For Kappa, these arealways near 0.2 for the parallelepiped classifier and 0.3 for the other two.

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4.9 - Parameters of the coherent target decomposition theorems 67

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It is to be noted that the improvement of the accuracy with increasing dimensionsof the averaging windows is now much faster than in the previous cases. Indeed, forthe Min. Dist. and Max. Lik. classifiers the overall accuracy results are already accept-able (K > 0.4) for the 3×3-pixel window and reaches values of the order of 0.75 for15×15-pixel window, which indicates an excellent classification performance. These arethe best results ever obtained with these two algorithms. Although the parallelepiped clas-sifier presents improved results too, it seems to provide basically the same accuracypercentages.

With reference to the largest averaging window, one obtains very good Us. Acc. andPr. Acc. for all the classes with all three algorithms. Some limits are presented onlyby the Parall. classifier with the classes “water”, “roads” and “grass”. Indeed, thisalgorithm completely fails in recognizing the training pixels of the class “water” andclassifies those of the “grass” only with an Us. Acc. of 6.33% and a Pr. Acc. of 0.46%.

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(c)

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The second TD method taken into account in our analysis is the Pauli decompo-sition. The tests performed on its characteristic parameters, the a, b and c complexcoefficient appearing in (2.120), do not reach as good accuracy values as those of theKrogager decomposition. The Ov. Acc. of the parallelepiped classifier remains almostconstant for all windows sizes and its values for the other algorithms do not differsignificantly from those obtained by simply using the amplitudes of the three [S] ma-trix elements. Hence, it seems that no meaningful improvement in terms of amountof information is provided by this TD theorem with respect to the original data (seeFigure 4.10).

4.9 - Parameters of the coherent target decomposition theorems 69

(a) (b)

(c)

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The detail of the single classes tells us that the best results in terms of producer’s anduser’s accuracy are those relative to the classes “houses” and “trees”, whereas “grass”,“water” and “roads” present much poorer values. Again, the parallelepiped algorithmfails to recognize the last three classes. As in the case of the tests directly performed onthe [S] matrix elements, one may suppose that the bad classification results for theseclasses are due to their low SNR that is not improved by the decomposition.

As a final example of the coherent TD theorems, we studied the Cameron decom-position [CL92], [CYL96]. As SAR data are calibrated in order to fulfill reciprocity con-straints, the basic distinction among scatterers made by this method is based on their


(a) (b)

(c)

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symmetry with respect to axes lying on a plane orthogonal to the radar line-of-sight.For this reason one has to perform only the decomposition (2.123) of the [S] matrix intoits most dominant and least dominant symmetric terms (henceforth max sym and minsym). The norm of the corresponding scattering vectors was then given as input to theclassifiers.

It is evident from Figure 4.11 that the accuracy estimations now reach values compa-rable with those obtained with the Krogager decomposition. Their maxima are againobtained for the largest averaging window; the Ov. Acc. ranges from 52% for the Par-all. classifier to 76% for the Max. Lik. one. For this window size, one can note how

4.10 - Entropy/α parameters 71

the maximum likelihood classifier yields good results even for those classes usuallymisclassified by the other algorithms, for instance “water” and “grass”. Indeed, theUs. Acc. remains above 50% in both cases, equal to 64% and 68% respectively. On thecontrary, the parallelepiped classifier fails again to classify the test samples belongingto these two training areas. Finally, it is also interesting that the Max. Lik. and the Min.Dist. classifiers provide the same accuracy levels for the pixel-based tests and for thoseperformed using the two smallest windows.

4.10 Entropy/α parameters

The entropy parameter was discussed in Paragraph 3.3. It represents the capability ofthe diagonalized coherency matrix [T(3)] to distinguish among the different scatteringmechanisms that contribute to the backscattered signal from a given resolution cell.As stated in that section, whilst the eigenvectors of [T(3)] each describe an independentscattering mechanism, its three eigenvalues (used to determine H) express their rela-tive intensity and hence give a measure of the “complexity” of the interactions in a cell.Further information may then be obtained by means of the parameterization given in(3.13): the type of the identified mechanisms is related to the α angles and a generaldescription of the given coherency matrix is related to the average α value.

Properties of the scatterers may then be interpreted in terms of these parametersand their classification performed considering their distribution in the H/α space (seeFigure 4.15). Since some physical limits exist for the variation of α as a function ofH , not all the H/α space represents real scatterers (i. e., not all the values of entropyand α have a physical meaning). Hence, one can define curves limiting this space and,inside this physically feasible area, make a further distinction between eight classes ofscattering mechanisms. These are clustered as follows [CP97]:

� � �� H/α

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� # �&�� ! � '"� � ��


Low entropy surface scattering. In this zone, low entropy scattering processes occurwith α values less then 42.5◦. All surface single scattering phenomena shouldbelong here: those within the geometrical optics limit, those treated by meansof physical optics, Bragg surface scattering and specular scattering phenomenawhich do not involve 180◦ phase inversion between Shh and Svv. For instance, atL-band, water and very smooth terrain surfaces belong to this category.

Low entropy dipole scattering. This area corresponds to strongly correlated mecha-nisms having a large imbalance between Shh and Svv in amplitude. Isolated di-pole scatterers, as well as vegetation having clearly oriented anisotropic scatter-ing elements would appear in this zone.

Low entropy multiple scattering. Double or even-bounce scattering events are to belocated here. This is the case for isolated dielectric and metallic dihedral scatter-ers.

Medium entropy dominant surface scattering. This zone presents a higher value ofentropy due to increases in surface roughness and to canopy propagation effects.Hence, surfaces characterized by an increasing roughness/correlation length orsurfaces covered by oblate spheroidal scatterers (such as certain types of leaves)would be located in this area.

Medium entropy dipole scattering. Moderate entropy with a dominant dipole scat-tering mechanism characterizes this zone that includes scattering from vegetatedsurfaces with anisotropic scatterers and moderate correlation for the orientationof the scatterers.

Medium entropy multiple scattering. Scattering phenomena contained in this regionare those typically present in forested areas; in effect, double bounce mechanismsthrough the canopy increase the entropy. Also urban areas belong to this zone,as dense packing of localized scattering centres can generate moderate entropywith low order multiple scattering being dominant.

High entropy vegetation scattering. Scattering phenomena from forest canopies liesin this region as does the scattering from some types of vegetated surfaces withrandom highly anisotropic scattering elements.

High entropy multiple scattering. In this zone, double bounce mechanisms in a highentropy environment may still be distinguished. Good examples are forests orscattering from vegetation that has a well-developed branch and crown structure.

The proposed division of theH/α space may not only be directly applied to performunsupervised classification but, as well, considered as a tool to check the accuracyof other classification techniques and their consistency. This is the strategy adoptedin this work. Again, the three chosen supervised classification algorithms were usedto perform tests with different sizes of the averaging window; in this way, a directcomparison with the other parameters is possible [ACP02b].

The values of H , α and A are calculated after the averaging process, indeed, thisis performed when evaluating the coherency matrix. As in the previous cases, the


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averaging windows have the following dimensions: 3×3, 5×5, 10×10 and 15×15pixels. A first series of experiments is based only on the H and α parameters, i. e., onlythese two have been given as input to the classifiers. The results of the three differentmethods in terms of Ov. Acc. and Kappa are illustrated in Figures 4.16 (a) and (b).A second series also takes into account the anisotropy A (hence, three parameters asinput). The classification tests were repeated for all the cases and further graphs of theOv. Acc. and Kappa were derived (see Figures 4.17 (a) and (b)). Tables summarizingthe main results of the classification tests can be found in Appendix C.4.

In both series of tests, the level of accuracy remains relatively low. Because of this,in [SAC+00] the total backscattered power was used as temporary additional input forimproving the segmentation of the H/α parameters.

The Ov. Acc. and the Kappa coefficient diagrams show again an increasing accuracyof the classification results ranging from the 3×3 to the 15×15-pixel window. The twosimplest classifiers, namely the Parall. and the Min. Dist., do not present remarkable

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accuracy values. They are not sensitive to the additional anisotropy information: thevalues of the overall accuracy and of Kappa are not subject to significant variations. Inparticular, the Parall. algorithm provides similar results independently upon use of theanisotropy as input. Also the Max. Lik. appears to be not so effective as it was whenapplied to the SDH coefficients with averaging, even though an increase of accuracyof 10% has often occurred when adding anisotropy. Moreover, the Gaussian statisti-cal hypothesis characterizing the Max. Lik. criteria is not satisfied by the polarimetricvariables under study and, as stated before, represents a potential flaw in the use ofthis classification method.


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A visual analysis of the resulting classification maps also suggests some limitationsof this method. The similar behaviour of the “water” and “roads” classes makes itimpossible to distinguish them from each other. The urban area appears well defined,but cultivated fields are often assigned to “houses” or “water” classes. Different kindsof fields are not separable. Only the class “trees” shows good classification results andthe measured values characterize it unambiguously.


4.11 Comparisons and conclusions

It is now possible to summarize the results discussed in the previous sections referringalso to Table 4.3 in order to get a general overview over them. Since the tests wereperformed on a single data set, the considerations that will follow do not presumeto have an absolute validity but should simply indicate a trend that is reasonable toassume valid for similar data and imaged scenes.

The first important observation is that little improvement in our classification tests canbe obtained using the incoherent polarimetric parameters derived from the original data ratherthan using directly these ones [PAC+02], [ACP02b]. Indeed, the Ov. Acc. increases justslightly for the tests on the H/α values with respect to those on the amplitude ones. Apossible interpretation is the following: the entropy and anisotropy parameters are de-rived by the eigenvalues of the coherency matrix; no use is made of the correspondingeigenvectors. One may then suppose that part of the information originally containedin the data has been “neglected” leading to poorer, or poorer than expected, classifica-tion results. Moreover, α is the only quantity that does not express an intensity.

Concerning the coherent parameters, only two of the decomposition methods, na-mely, the SDH and the Cameron decompositions, lead to significantly higher accuracyvalues, whereas the ratios of the [S] matrix elements [AC00] provide results signifi-cantly worse than all the other polarimetric parameters.

As a further note of caution, it should be reminded that the chosen algorithms im-plement quite general image classification methods and are not specifically intendedfor SAR data; hence, they are not the optimal tools for analyzing them. In general,the Max. Lik. classifier gives the best results (in terms of Ov. Acc. and Kappa coeffi-cient); independently from the averaging windows dimensions, the values obtainedare higher than in the other two cases. This may be attributed to the higher versatil-ity of the second order statistic model of the Max. Lik. classifier in comparison withthe first order statistics criteria adopted by the Parall. and Min. Dist. methods. Dueto these considerations, the unsupervised segmentation proposed in [CP97] would bemore appropriate for the incoherent parameters.

As expected, the choice of the training data set plays a relevant role: some of theselected ground coverage types always give rise to problems in the classification pro-cesses and bad accuracy results, whereas, for others, these are constantly acceptable.

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Class. alg. → Maximum likelihood Minimum distance ParallelepipedPol. parameters ↓ Ov. Acc. (%) Kappa Ov. Acc. (%) Kappa Ov. Acc. (%) KappaBackscattered amplitude 57.03 0.5 50.96 0.43 33.88 0.24Co- and cross-polar nulls 66.24 0.6 64.41 0.58 44.61 0.36SDH coefficients 87.37 0.85 77.04 0.73 55.58 0.48Pauli coefficients 57.61 0.51 52.27 0.44 34.65 0.25Cameron terms 76.37 0.72 71.35 0.66 52.9 0.45H/α 60.25 0.54 45.3 0.36 25.48 0.136H , α and A 64.1 0.58 45.31 0.36 25.71 0.137

4.11 - Comparisons and conclusions 77

In fact, in all the tests, the classes “houses”, “trees” and “field 1” were better recog-nized than the classes “roads”, “grass” and “water”. Hence, the role of the analyst witha ground truth knowledge remains fundamental, in the sense that the selection of the trainingdata and of the classification items affects the final results sometimes as deeply as the choice ofthe type of the inputs.

Regarding the comparison among single-pixel and averaged areas classifications,we have shown that, by averaging the values of the image pixels, the classification efficiencyis, in general, enhanced. This could mean that the studied parameters refer to physical prop-erties “spread” over neighbouring and somehow correlated pixels. Indeed, the presence of“extended” parameter characteristics may have led to improved classification accuracyfor increasing sizes of the averaging windows (at least in the limits of the 15×15-pixelwindow as largest size). Another aspect to be taken into account, which influences thedifferent results, is that the tests compare averages of both real and complex quantities.

It is worthwhile to stress that, to our knowledge, no systematic investigations simi-lar to ours have been yet conducted on this subject.

5 Analysis of polarimetric parameters:interferometry

5.1 Overview

Polarimetry and interferometry seem to play distinct roles in radar remote sensing,leading also to their application to different tasks: polarimetric analysis makes it possi-ble to separate different scattering mechanisms within a given resolution cell, whereasinterferometric techniques allow for a topographical characterization of the scatter-ing contributions. As an obvious consequence of these properties, one may try to usepolarimetry and SAR interferometry for those cases when different scatterers are ex-pected to occupy different positions. In particular, a topic actively investigated is thepossibility to distinguish scatterers separated in height, and steps in this direction havebeen recently taken [CP98], [PRC99a], [PRC99b], [PC01], [IC01], [Sag00]. The interestin doing this is readily found when considering the various sources of decorrelationaffecting two SAR images: a major cause is the volume scattering term, which is re-lated to the height distribution of the effective scatterers above the chosen referenceplane [ZV92], [BH98]. Hence, “resolving” the volume decorrelation and fixing the ex-act position of the scattering centres could also lead to an estimation of the height ofthe resulting distributed scatterers. Moreover, the exact determination of the volumescattering contribution is fundamental in DEM generation as well as in biomass esti-mation.

Due to all these reasons, the question of volume decorrelation is worth special atten-tion. We present here a study on the potential of target decomposition methods com-bined with interferometry which is aimed at verifying if they can be used to reducethis effect and the problems connected to this approach. Given the differences amongTD theorems, in particular among coherent and incoherent ones, we also investigateif different methods permit us to distinguish between volume scattering (that is, veg-etated/forested areas) and coherent targets with a profile along their height (basicallybuildings and other man-made artifacts).

5.2 Theoretical aspects

In order to understand the phenomenon of decorrelation in SAR interferometry, it ishelpful to provide more details about the expression of the interferometric coherenceintroduced in Equation (3.20).

79

80 Chapter 5 - Analysis of polarimetric parameters: interferometry

5.2.1 Interferograms generation

Let us consider first a single SAR image and the geometry associated with it as given inFigure 5.1. Here, the primed coordinates are related to the target, while the unprimedones refer to a reference point for the system response h(x,R). With this geometry, thevalue of the image for a point on the ground at x′ azimuth position and slant range R′

can be expressed as [HUA95], [BH98]:

s(x,R) =

∫

u(~r′) exp(−j2kR′)h(x− x′, R−R′)dV ′ + n(x,R). (5.1)

In (5.1), ~r′ = (x′, y′, z′) and dV ′ = dx′dy′dz′, u(~r′) represents the three-dimensionalcomplex terrain reflectivity (it will be referred to also as the scattering object), h(x −x′, R−R′) and n(x,R) are respectively the system response and noise, and ~k the signalwavevector which results as function of the radar look angle θ:

~k = |~k|(0, sin θ, − cos θ). (5.2)

The system response is usually modulated by rectangular filter functions of bandwidthWR in range and Wx in azimuth, and hence has the form:

h(x,R) = sinc(Wxx) sinc(WRR) =sin(Wxx)

Wxx

sin(WRR)

WRR. (5.3)

��

��

’

x−x’

θR(R’, x−x’)

V’

H

sy

z

y

xx’

z’

R’

y’

�� &� �� ,�!�'�� R(R′, x − x′)

�� ! ��&� ' � �0#$!�� %"�� '��&� '"� � � !�' � �� '� �� !��

5.2 - Theoretical aspects 81

��

y

ξ

H

z

ys

R sθ

η� � �&�� ,�!�'�� # ��&�� &� '"!�� . � �&��! � � � ��&� ' � � ' ! �� .��

Now, let us further assume that the origin of the range coordinate in the SAR imagehas been set to:

Rs =√

H2 + y2s (5.4)

so that we can define a new range coordinate η as:

η = R−Rs (5.5)

and an axis ξ orthogonal to the η-axis directed as represented in Figure 5.2. In this newcoordinate system, it is possible to derive the plane wave approximation of (5.1) as:

s(x, η) = exp(−j2kRs)

∫

u(~r′) exp(−j2~k · ~r′)h(x− x′, η − η′)dV ′ + n(x, η). (5.6)

When doing interferometry (see Figure 5.3), one works with two images character-ized by different objects u1(~r1) and u2(~r2) which pass system responses h1(x, η) andh2(x, η) according to Equation (5.6). Due to the slightly different look angles, the (η, ξ)-coordinate systems of the images differ. However, the difference ∆θ between the lookangles is small enough such that an average angle θ0 and a single (η, ξ)-coordinate sys-tem can be considered. Hence, only the exponential factor exp(−j2~k · ~r′) in (5.6) isdifferent in the two wave-vectors.

As the two SAR images may not have been acquired simultaneously, the scatter-ers may have changed between acquisitions (we will see later the possible causes ofthese changes) and the two complex reflectivities are connected by a cross-correlationfunction such that:

E[u1(~r1) u∗2(~r2)] = σve(~r1) δ(~r1 − ~r2), (5.7)


SB

B

P

θ

S

RB

R

z

y

� ��

� ��

� ��

� ��

∆θ

θp 1n

2

1

2

1

2

� � �&�� &' � �� ' � �� &. �� 0� # �!�� !�� ! �� &. ��

where E[ ] stands for expectation value and σve(~r) is the volumetric backscatter coefficient[BH98] of scatterers common to both images and expresses the amount of correlationof the two complex reflectivities. The coefficient σve(~r) can be interpreted as the tem-porarily stable scattering contribution; scatterer contributions that have changed be-tween observations average out in the expectation value.1

Assuming mutually uncorrelated system noise n1(x, η) and n2(x, η) of intensity:

E[|n1|2] = N1 (5.8)

andE[|n2|2] = N2 , (5.9)

one may use the equations above to write the interferometric coherence as:

γ= |γ′|= |∫

σve(~r′)h1(x− x′, η − η′)h2(x− x′, η − η′) exp[−j2(~k1 − ~k2)·~r′]dV ′|√

(S1 +N1)(S2 +N2)(5.10)

and take the phase of the complex γ ′ as the expected interferometric phase.

S1 and S2 are the noise-free signal intensities of the two images and are equal to:

S1 =

∫

σv1(~r′)|h1(x− x′, η − η′)|2dV ′, (5.11)

1This is true only for random changes of scatterers. Hence, this form of cross-correlation function isnot applicable for objects that perform a rigid movement and whose scattering properties remain stable.


S2 =

∫

σv2(~r′)|h2(x− x′, η − η′)|2dV ′, (5.12)

where σv1(~r′) and σv2(~r′) represent the volumetric backscatter coefficients of the scat-tering objects and are a measure of their autocorrelation.

5.2.2 Decorrelation sources

According to [RM92] and [ZV92], the coherence can be written as the product of severalcontributions:

γ = γSNR γtemporal γspatial. (5.13)

γSNR represents the decorrelation due to additive noise, while γtemporal stands for tem-poral scene coherence and is defined as the ratio of temporarily stable scattering con-tributions to the total scattering intensity transferred to the SAR images (more detailsabout these contributions and their analytical formulation can be found in [ZV92],[BH98] and [Pap99]).

The contribution deserving our attention in this chapter is γspatial , which describesthe decorrelation caused by the different processing performed on the two SAR sig-nals (in the sense that they have been differently “coupled” to the system responses).In turn, spatial decorrelation itself can be split into two terms relating to different scat-tering mechanisms, surface scattering and volume scattering, that is:

γspatial = γsur γvol . (5.14)

This is possible under the assumption:

R2 = x2 + y2 + z2(x, y) ≈ x2 + y2 (5.15)

which, referring to Figure 5.1, implies the possibility of choosing the origin of the co-ordinates on the mean plane of the surface and far from the illuminated area. Hence,it holds: x, y � z and the dependence of h(x, η) from z can be neglected.

The characteristics of the targets on the ground and their differences (in size, shape,distribution and temporal stability) deeply influence the SAR image synthesis and theinterferometric application [BH98]. Indeed, due to the high sensitivity to range varia-tions of the phase term in the integral of (5.6), each scatterer in a resolution cell (meters)should be located with a precision of the order of a fraction of the wavelength (centime-ters). This condition is hardly met for natural scenes that usually contain distributedtargets. Also the expression of the volumetric backscatter coefficient depends com-pletely on the nature of the imaged targets and two extreme situations are possible:point scatterers and Gaussian (or Rayleigh) scatterers.

The first case relates to ideal targets whose volumetric backscatter coefficient is ofthe type:

σve(~r) = σ0 δ(~r − ~r0). (5.16)

This means that the resolution cell may be treated as point-like at least concerning thedetermination of its position. Hence, the coherence is equal to 1 and the interferometricphase is exactly that which corresponds to the path difference from the two antennaeto the target with no uncertainties in range. Such a case is realized, for instance, by “ad


hoc” artificial targets, the so-called corner reflectors, whose backscatter is so strong as tocompletely determine the value of σve(~r) for that resolution cell since the contributionfrom surrounding natural targets is overcome. Backscattering from Gaussian scatterersis, on the contrary, due to several elementary random scatterers among which noneprovides a contribution clearly dominating the others.

Another aspect to be considered is the actual spatial distribution of the elementaryscatterers, in particular, their eventual displacement in height. For example, the canopylayer of forests is constituted by randomly distributed scatterers, the leaves and thebranches, with varying shape and orientation which are also subject to frequent, andrandom, changes in time. Backscattering from such targets usually yields very lowcoherence values and is referred to as random volume scattering. It is important to stressthat the low coherence is due both to the vertical, almost continuous, structure of thetarget (for reasons that will be explained below) and to its low stationarity. For the casewhen only varying target heights affect the coherence but the elementary scatterers arestationary, discrete and limited in number, as those of walls or buildings, then it ishelpful to introduce the new definition of coherent volume scattering [ACP02a], [AC03].

Coming back to Equation (5.14), one can consider the two contributions separately.The case of pure surface scattering is characterized by:

σve(~r) = σsur(x, y) δ(z − z0), (5.17)

that is, all scattering interactions are assumed to take place at a certain height z = z0

and the degree of correlation depends on the phase variations of the scatterers on thisplane. As we will see, it is possible to compensate the effect of the decorrelation dueto changes in look angle of the sensors, by means of an adequate setting of the systemresponses.

Quite different is the scenario when volume scatterers are present. After compen-sating pure surface effects, γspatial is determined by a volumetric backscatter coefficientwith a profile in z and constant in x and y:

σve(~r) = σvol(z). (5.18)

Its characteristics change significantly depending on whether it represents coherentvolume scatterers (buildings or other man-made artifacts) or a random volume (vege-tated areas), though for both γvol has the form:

γvol =|∫

σvol(z′) exp[−j2(kz′,1 − kz′,2)z

′]dz′|∫

σvol(z′)dz′, (5.19)

where the components kz are the projections onto the z-axis of the wavevectors. In-deed, in the case of coherent volume scatterers, a limited number of elementary scatter-ers can be found in each resolution cell or, at least, the number of multiple interactionsis less than in a random volume so that the coherence tends to values nearer to 1 thanin the second case.

5.2.3 Interferometric coherence enhancement

The reduction of the decorrelation effects of γspatial follows different strategies accord-ing to the mechanisms originating γsur and γvol. Under the hypothesis of pure surface


scattering, the decorrelation is due only to the fact that the backscattered signals corre-spond to different bands of the ground reflectivity spectrum [PRGP94], [GGP+94]. Thiscan be demonstrated starting from the approximated relation between the frequency fand the ground range wavenumber ky:

ky = 22π

λsin(θ − α) =

4πf

csin(θ − α), (5.20)

taking into account the two-way travel path and also a constant uniform slope of theterrain represented by α. The difference ∆ky, i. e., the variation of ky generated by aslight change of the look angle ∆θ, is then:

∆ky =4πf∆θ

ccos(θ − α). (5.21)

Thus, in general, a look angle difference ∆θ generates a shift and a stretch of the imagedterrain spectra. However, if the relative system bandwidth is small, the frequency f in(5.21) can be substituted by the central frequency f0 and the stretch can be neglected,then the following equation holds:

∆ky =4πf0∆θ

ccos(θ − α). (5.22)

Finally, since the radar is not monochromatic (a certain bandwidth W centered aroundf0 is always given), one may conclude that by changing the look angle of the SAR beamone gets a different band of the ground reflectivity spectrum. This difference is a sourceof decorrelation that affects γspatial but, as was shown in [GGP+94], it can be overcomeby means of tunable radars, i. e., systems whose processing filters can be tuned to thedifferent centre frequencies (a technique known as wavenumber shift filtering):

h2(x, η) = h1(x, η) exp(−j2π∆fη), (5.23)

where:∆f = − 2Bn

λRs tan(θ − α)(5.24)

and Bn is the component of the baseline normal to the look direction [BH98]. Equa-tion (5.24) does not relate to angular dependence of the scatterers, an assumption validin this case.

A first method for resolving volume decorrelation by means of polarimetry wasproposed by Cloude and Papathanassiou [CP98]. It consisted of associating a differ-ent scattering mechanism, and connected to that a scattering centre, to each scatteringvector defined by the optimization of the interferometric coherence (see page 45). Ac-cording to their expectations, the maximization constraints would act mainly on thevolume term of γspatial (in the hypothesis of reduced effects of temporal decorrelation).Moreover, in this way, the position in height of the scattering centres can also be de-fined. Thus this method should provide a first rough estimate of the height of extendedtargets.

In [CP98] and [Pap99], this result was suggested as a way to measure tree heights,assuming that the scattering centres so identified are situated almost at the top and at


z=h

z

z=0

V

�� &� �� )�# �"��. !�� # � �� ' � !� � �&' �� !�'� ��&. � � � � . � �"��&� �&��'� �(# !�� 0� '�� . � �� &�� ' � � � � ��

the base of the trees. Unfortunately, when applied to the estimation of forest heights,this direct approach presents limits that depend on the properties of the different scat-tering components. The main point is that, while scattering from a random volume(that of the branches and leaves layer) is unaffected by changes of polarization, theground scattering is polarization-dependent but, since no polarization contains onlyone of these two contributions, it is not possible to completely separate them. As aresult, the straightforward evaluation of their difference in height through their inter-ferometric phase difference leads to underestimated values.

A different approach has been adopted specifically for the case of vegetated areas[PRC99a], [PRC99b], [PC01], [Sag00]. It is based on the use of scattering models con-necting the interferometric measurements to some basic parameters characterizing thevegetation, such as the height of the canopies and foliage volume and its extinctioncoefficient. By means of these limited sets of parameters, simplified expressions of in-terferometric quantities (usually the coherence) can be derived; then, via minimizationprocedures of the difference between modelled and measured interferometric values,it is possible to invert the experimental measures and obtain an estimate of the chosenvegetation parameters. The main difficulty of this way of proceeding is the formula-tion of the models, which must be simple enough to allow the inversion but, at thesame time, be able to properly describe the complexity of the imaged vegetation layer.

In [PRC99a], [PRC99b] and [PC01], an example of such a model is presented basedon an interactions scenario which includes random volume as well as ground scatter-ing [TS00]. It consists of a volume of randomly oriented scatterers above the ground.This is a good approximation for scattering at L-band. With the chosen model, onecalculates the interferometric coherence taking into account direct backscattering fromthe ground and from the random volume; the contribution of multiple bounces is ne-glected. The parameters related to the vegetation layer are its thickness hV , the scatter-ing amplitude mV (w) for a given polarization w of the randomly oriented scatterers,


and the volume extinction coefficient σ (in dB/m). Related to the ground beneath thevegetation are the topographic phase φ0 and the scattering amplitude mG(w). Usingagain a vertical axis z with the origin on the ground surface and the same geomet-ric parameters (average radar look angle θ0, range distance R, baseline B) introducedbefore, the complex interferometric coherence is written as [PC01]:

γ′(w) = ejφ0γ′V +m(w)

1 +m(w), (5.25)

with m(w) representing the effective ground-to-volume amplitude ratio:

m(w) =mG(w)

mV (w)exp

(

−2σhV

cos θ0

)

(5.26)

and γ′V the complex coherence for the volume alone given by:

γ′V =

∫ hV

0exp

(

2σzcos θ0

)

exp(jkzz)dz

∫ hV

0exp

(

2σzcos θ0

)

dz. (5.27)

kz is the effective vertical interferometric wavenumber after spectral shift filteringand is equal to:

kz =kB cos θ0

R sin θ0

. (5.28)

The volume extinction coefficient σ represent a mean extinction value of the vegetationand depends on the scatterers density and on their dielectric constant.

It is interesting to note that, due to (5.25), the position of the effective scatteringcentre is defined so that it lies above the ground at a height defined by the ground-to-volume amplitude ratio and by the attenuation length of the vegetation.

By adopting the same model of the vegetation and of the underlying ground, Sagues[Sag00] derived an alternative expression for γ ′(w) after spectral filtering:

γ′(w) =ejφ0

K(w)

[

eχhV − 1

χ+m(w)hV

]

, (5.29)

with:

K(w) =

[

e2σ cos θ0hV − 1

2σ cos θ0

+m(w)hV

]

(5.30)

and

χ = 2

[

σ cos θ0 cos(∆θ/2) − jk0∆θ

sin θ0

]

, (5.31)

where k0 is the wavenumber corresponding to the central frequency f0.

To invert the models, the complex interferometric coherence of at least three differ-ent polarizations must be available so that one has a number of observables equal tothe number of parameters involved. Hence, one must calculate the interferometric co-herence using different polarizations. In this sense, a useful contribution could comefrom target decomposition theorems, especially if they could provide a way to better


define the ground-to-volume amplitude ratio or to distinguish volume scattering dueto random volumes (for which the models were built) from scattering due to coherenttargets extended in height.

The following sections contain a description of some TD methods analyzed andcompared with the aim of establishing whether some of them could be helpful forsuch problems and, if this is not the case, for suggesting more appropriate fields ofapplication.

5.3 Interferometric coherence analysis

In this section we address the analysis of the correlation properties of the scatteringmechanisms contemplated in a series of decomposition theorems. We will describe atfirst their general characteristics and then we will go more deeply through the questionof the volume decorrelation, its causes and its possible reduction [AC01], [ACK02],[ACP02a] and [AC03].

The study was conducted again with L-band data of the area of Oberpfaffenhofen,Germany, acquired by the E-SAR sensor of DLR during two different measurementcampaigns (May ‘98 and October ‘99). Both sets are suitable for interferometry, withbaselines of 15 m and 12 m respectively. For the coherence images, an averaging win-dow of 6×12 pixel (in range and azimuth, respectively) was adopted.

5.3.1 General correlation properties

Let us start by analyzing the SDH decomposition [Kro93] and indicate its generic ithterm as:

[S]i =

[

ShhiShvi

SvhiSvvi

]

, (5.32)

where i refers each time to sphere, diplane or helix. With this technique, the interfero-metric coherence may be derived in two different ways: using the hh-elements of thethree matrices and also by means of the corresponding scattering vectors. In the firstcase, for each scattering mechanism, one has to simply calculate [AC01]:

γi =|〈Shhi,1

S∗hhi,2

〉|√

〈Shhi,1S∗

hhi,1〉〈Shhi,2

S∗hhi,2

〉, (5.33)

In Figures 5.5 (a), (b) and (c), the results relative to the data set of May ‘98 are repre-sented; Figure 5.6 shows in more detail the interferometric coherence of the Shh ele-ments of the diplane-like matrix for the second data set.

When using the scattering vectors, Equation (3.34) can be applied directly and, forboth images of the interferometric pair, one projects the original scattering vectors onthe unitary vectors representing a given scattering mechanism. For example, for thefirst image of the pair and the ith term of its decomposition:

µ1i= w

†1ik1 , (5.34)

5.3 - Interferometric coherence analysis 89

(a) (d)

(b) (e)

(c) (f)

� � �� ' � �� . �� 0� # #$� �"��'"#$� ��(� �� /�$� �&.�� ! � ��Shh

�� . ��' � ��"� �� 0. � �%��

Shh�� . ��' � � �"� �� &� � � !�'�� (. � ��# ��

Shh�� . ��' � � �� 0. � � �

�� '�� !�� !��0� � !�� &' ��# � �&� � � �� ' � � '�� -� �� (. � � � � �� '�� !�� !��(� � !��$� ��'��# � �&� � � �� ' � � '�� &� � � !�'�� 0. � �� '�� !�� !��0� � !�� &' � ��# � �&� � � �� ' �$� '�� 0. � �/!�� %�! � �� '"��& � .��(�

where:w1i

=k1i

‖k1i‖ (5.35)


�� &� �� ' � �� . �� 0� # #$� ��' #$� ��(� �� $��. ��Shh

�� . ��' � �� &� � � !�'"� � ��(.� � # � �&%�� %�!��(�� '��& � .��

and k1iis calculated using the Pauli basis:

k1i=

1√2

Shhi,1 + Svvi,1

Shhi,1 − Svvi,1

2Shvi,1

. (5.36)

As stated before, the final results do not depend on the chosen basis. Of course, theconstraint (3.39) on the phase of the w vectors of the two images corresponding to thesame scattering mechanism must be taken into account.

As a first comment, it should be noted in the images and graphs reported here thatthere are some differences between the two series of coherence images, one based onthe hh-elements and the other on the coefficients of the unitary component matrices.The latter should be preferred, since they seem to be less sensitive to noise effects thanthe hh-elements alone.

Considering the data as a whole, one sees that the three scattering mechanismspresent different correlation properties and that, in general, a higher correlation charac-terizes the sphere-like component with respect to the other terms of the decompositionand also to the corresponding non-decomposed data. This can be easily seen in thegraphs of Figure 5.9 that report the histograms of the coherence in the various cases.

The higher correlation of the sphere-like terms is particularly evident in the flat areaby the runway, but it is present also in the forested areas. Here, assuming the be-haviour of the canopies to resemble that of a random volume, the variation of the


coherence among the three images may indicate the presence of at least two contribu-tions due to ground scattering [PRC99a], [PRC99b], [PC01]. The degree of coherence ofthe diplane-like terms improves in the village and in a small and isolated forest standdirectly to the right of the airfield (see Figure 5.6). In both cases it is again the reducedeffect of the volume decorrelation which explains this result. Indeed, in urban areas,structures such as flat surfaces and walls are often present and they behave almost likeideal double bounce scatterers characterized by coherence values almost equal to 1.Generally, in the presence of man-made artifacts, the contribution of scattering fromisolated points at different heights decreases in comparison with that of mechanismsthat better resemble point scatterers. Quite interesting is also the scenario for the iso-lated group of trees: they cause a double bounce between the bare field around themand their trunks, hence another strong contribution of ground scattering with a well-defined mechanism.

The second TD method considered is that based on the Pauli matrices [ACK02]. InEquation (2.120), the three terms of the decomposition will be indicated as a, b andc or, respectively, as first, second and third term. Also in this case the three scatter-ing mechanisms show different correlation properties, with a (corresponding to thesphere-like term of the SDH decomposition) having the highest degree of coherence(see Figures 5.7 (a), (b) and (c)). This depends again on the stronger backscatteredsignals from the ground. In the forested areas of the scene, assuming again that the ca-nopies resemble a random volume, it is possible to observe variations of the coherenceamong the three images, suggesting different contributions of the scattering from theground beneath the canopies [PRC99a], [PRC99b], [PC01].

The behaviour of the overall coherence histograms shows only one relevant differ-ence between the two TD methods: the third terms of the Pauli decomposition have ahigher correlation degree than the helix-like terms of the Krogager one. Both elementsof the two decompositions are representative of depolarization processes occurringwith the scattering but, as they do not give similar responses, they seem to be able todistinguish different depolarization mechanisms.

We studied, as last example of coherent TD methods, the Cameron decomposition[CL92], [CYL96]. In fact, as SAR data are calibrated in order to fulfill reciprocity con-straints, the basic distinction among scatterers is based on their symmetry with respectto the axes lying on a plane orthogonal to the radar line-of-sight. This means that wehad only to perform the decomposition (2.123) of the [S] matrix into its most domi-nant and least dominant symmetric terms (max sym and min sym). Another point toremember is that, according to the theory, these two terms are orthogonal.

The interferometric coherences of the two components present aspects which arequite original with respect to those previously seen. Let us analyze Figures 5.7 (d) and(e) and Figure 5.9 (d); no other technique leads to such a separation of the coherencevalues among the various terms: the correlation of the max sym term is higher thanany of the decompositions terms already considered, while it is also notable the lowbackscattered intensity that leads the min sym component to have a very low coher-ence. Moreover, this enhanced separation characterizes the data as a whole, regardlessof the particular homogeneous subsets of land coverages.

For a better understanding of the results presented to now, we compared them witha series of images obtained from the optimal polarizations which maximize the inter-


(a) (d)

(b) (e)

(c)

�� &� �� ' � �� &. �� 0� # # � �� '"# � ��(� �� $� �&.�� ! � �� !�� (. ��%�� ' � !�� (. ��# � ��"� �� 0. �� "�/�(# !�� (� '�� # � �&� � �� !�. �� &' . � � � ��&.�� '�!�' � � ��.

�. ��$�(� # � ��(. � � � � ��0#$!�� (� '��/��# � �&� �� !�. �� &' � � ! � � ��&.�� '�!�' � ��.�. ��$�(� # � ��0. � � !�� %�!��(�� '�� . �(�

ferometric coherence [CP98], [Pap99]. These polarizations are no longer related to acoherent decomposition as the coherence optimization does not operate on the [S] ma-trix. For obtaining these images we used an averaging window of 6×12 pixels; hence,


(a) (d)

(b) (e)

(c) (f)

� � �� ' � �� &. ��$�(� #�#$� ��' #$� ��0� � � � ��./� � ! � ��"� �� &� �$� . !�� !�� %�� '� �&� � � . !��$� !�� # �� &� �$� . !�� !�� &�0� �&� '�!��

Shh�� . ��' � � � � � �� (� �&� '"!��

Shv�� . ��' � � � � � �� &�(� �&� '�!��Svv

�� . ��' � � � �/!�� %"!�� '��& � . �(�

the results are comparable with the previous coherent ones.

When considering the actual improvement of the coherence, one sees in Figure 5.8that only the first optimal polarizations are highly correlated. The other two polariza-tion pairs present a degree of coherence comparable to that of the terms of the previous


(a) (b)

(c) (d)

(e) (f)�� &� �� &�!�. � �� "� � ' � �� &. ��$�(� # #$� ��'"# � �"� ��-� �� !�� ! ��0� � � *� ��&.�� ! ��

Shh�� "� )�� #$�&.�� &' � ��(.*� � � %�� 0#$!�� 0� '"�*��# � �� "� �� )�� #$�&.

��"� � � � � �&' � ��(.*� � � # � �� !�� # �&.�� $� �&'�#$�� #�� ' � � � � � �� 0#$!�� 0� '"�/��# � �&� ��"� ��"�� !�. ��' ��#$��./�"� � � � � �&' � ��(.*� � � � ��"� �&� � � . !��"� � !��(� � !��$� �&' � � �� "� �&�(� �&� '�!��"� � !��(� . �� 0� # !�� !�� /!�� %�!��(�� '��& � .��

TD techniques.

The optimized coherence images provide a valuable comparison with the others,in particular when trying to estimate how well a coherent decomposition theorem isable to identify and separate targets with given deterministic properties, i. e., to distin-guish those targets with a relatively high degree of coherence but whose simultaneouspresence in the very same resolution cell leads to a reduction of its total value. More-


over, they should also be able to describe distributed targets by finding their “bestrepresentation” as ideal point scatterers. This property of the optimization methoddistinguishes it from the coherent ones whose application to distributed targets is stillcontroversial. Due to these facts, one should expect similar behaviours of the coherentand incoherent methods in presence of deterministic scatterers whereas they shouldbehave differently when dealing with partial ones. Practically this can be seen at thepixel level by comparing the optimal coherences to those relative to the various mech-anisms of another method and seeing also when they correspond to each other.

Remarking again that targets with symmetry characteristics are also the most co-herent, at least according to the results obtained with the Cameron decomposition, itseems conceivable to suppose that the first optimal polarization always identifies thosetargets having this kind of property.

Some coherence images relative to the original acquired data have also been in-cluded for comparison; they are reported in Figures 5.8 (d), (e) and (f). The main infor-mation that can be retrieved from them regards the predominance of Bragg or Fresnelscattering from the ground [CM91], [BW85], [UMF82]. The imbalance in backscatteredintensity between the hh- and vv-channels yields also variations of the coherence val-ues, so that a correlation of the Shh elements higher than those of Svv means a preva-lence of Fresnel type scattering, typical of almost flat surfaces, whereas the oppositecase is verified for scattering from rough surfaces described by the Bragg model.

5.3.2 Volume decorrelation

As is well known, buildings, houses and other constructions cause volume decorre-lation just as trees do, but while for random volumes some models for the interfero-metric coherence are given [TMMvZ96], [TS00], [PC01], to our knowledge, none hasbeen presented for “coherent” extended targets. In fact, due to their deep intrinsicdifferences, urban areas and forests lead to decorrelation effects having original char-acteristics each and, hence, require different coherence models. However, even whenprecise models are not available, it is interesting to see whether these differences can atleast be enhanced and recognized by means of TD theorems. Let us consider again thecorrelation properties of the terms in each decomposition procedure, limiting our anal-ysis to forested and urban areas (we selected areas whose expanse, in terms of numberof pixels, is almost identical).

For each mechanism of the various methods, there is evidence of different behaviourof the coherence in forests and towns; more in detail, the decompositions show thatthere exist different sources of volume decorrelation and that, for reasons we will ex-plain later on, this can be better resolved when it is due to targets resembling idealpoint scatterers. Comparing Figures 5.10 and 5.11 it is evident that the urban area hasa much higher coherence than the forest. In both cases, no meaningful improvementseems to be given by the Krogager and Pauli decompositions in general and only themax sym Cameron terms and the first optimal polarizations show an enhanced coher-ence [ACP02a], [AC03].

Regarding the coherence distributions, one may note this: for the forest, the his-tograms of the hh, hv and vv-channels are basically identical and so are also those of


(a) (b)

(c) (d)

(e) (f)�� &� �� &�!�.*� �� /� ' � �� &. ��$�(� # #$� ��'"# � � �&� ��0%�!�'�!�� ! � ��0� � � �� &.�� ! ��

Shh�� "� )�� #$�&.�� &' � ��(.*� � � %�� 0#$!�� 0� '"�*��# � �� "� �� )�� #$�&.

��"� � � � � �&' � ��(.*� � � # � �� !�� # �&.�� $� �&'�#$�� #�� ' � � � � � �� 0#$!�� 0� '"�/��# � �&� ��"� ��"�� !�. ��' ��#$��./�"� � � � � �&' � ��(.*� � � � ��"� �&� � � . !��"� � !��(� � !��$� �&' � � �� "� �&�(� �&� '�!��"� � !��(� . �� 0� # !�� !�� /!�� %�!��(�� '��& � .��

the Pauli terms. The corresponding histograms for the town data present shapes whereonly small differences among the decomposition terms are recognizable. In general,


(a) (b)

(c) (d)

(e) (f)� � �� &�!�. ��"�� ' � �� &. ��$�(� # #$� �"��'"#$�� &� �� !��!�� 0� �� ./�� ! ��"�

Shh�� )�� #$�&.�� &' � ��(.*� � � %�� 0#$!�� (� '��*�� # � �� )�� #$��.

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this can be explained assuming decorrelation mechanisms that are polarization inde-pendent; in this case, indeed, similar values of the coherence, distributed in the same


way, should be expected. On the contrary, if a scattering mechanism depends on thepolarization of the incident wave, so does its correlation and the question is then to findwhich polarization puts into evidence this dependence most clearly. According to thegraphs, only the Pauli decomposition partly fails in doing this; all the other methodslead to coherence distributions with an original behaviour for each term. In particular,the curves of the Cameron decomposition terms and of the optimal polarizations areclearly distinguished but their shape does not have not a special meaning.

The point of the shape of the correlation distributions is also worth attention andsome comments about it would be better understood by looking at the graphs of theurban areas. In an ideal case, given a single point scatterer for each resolution cell ofa scene, the coherence would always be 1 and its distribution simply a δ-distributioncentered there. The more the targets differ from point scatterers the wider is the spreadof the coherence distribution to values lower than 1.

When more than one target is present in each resolution cell, the question is which isdominant and how do they combine together to return a single backscattered intensityfor a pixel. Due to mutual interactions and differences among the responses, the coher-ence will be lower than 1 even in the presence of point scatterers if these are verticallydistributed. Only in the case of orthogonal (i. e., completely independent) scatteringmechanisms, is it possible to recover the individual responses separately by selectingthe appropriate polarizations and obtain again coherence values equal to 1. As TD the-orems perform this separation, one should expect that, for a resolution cell containingthe very same targets of the decomposition model, the coherence of each term wouldbe 1 or close to 1 depending on the orthogonality of the mechanisms. Hence, the shapeof the coherence histograms of the various terms will also tend to a δ-distribution cen-tered at 1. In principle, this could represent a measure of the correctness of the adoptedmodel: the departure of the coherence histogram from a δ-distribution should revealhow well a certain theoretical model explains the actual targets present in the scene.This is why we anticipated that coherent decompositions seem to better resolve volumedecorrelation when it is due to targets resembling ideal point scatterers (as it happensin urban areas) than in the presence of random volume structures (forests). Indeed,in both situations the experimental results indicate the presence of several scatteringmechanisms (as in the forested areas where different contributions from the groundare evident), but for the town data the histograms of the coherent decompositions re-semble those of the optimal polarizations and tend to δ(x − 1), whereas this trend isnot observed for the forest data.

5.4 Interferometric phase analysis

As demonstrated, the method adopted in [CP98] and [Pap99] to resolve volume decor-relation by means of the optimal polarizations shows some limits when dealing withrandom volumes (that is, the case of scattering from vegetated areas). The underes-timation of the tree heights with this technique is due to the fact that the orthogonalscattering mechanisms refer to scattering centres placed not exactly at the basis and onthe top of the trees. It is then interesting to investigate other polarizations and see ifbetter results can be obtained. Hence, we evaluated the expectation value of the in-

5.4 - Interferometric phase analysis 99

terferometric phases of our data in all of the considered cases of TD methods. Thisanalysis was conducted on a basic level, directly considering the interferograms andtheir differences without retrieving from them the corresponding height values.

Let us start by considering again the SDH decomposition. Since it is not derivedfrom a basis of matrices (as, for example, the Pauli ones), the resulting terms are notmutually orthogonal; nevertheless, at least one matrix of the Krogager decompositionis orthogonal to the other two, so that one may assume the corresponding scatteringmechanisms to be statistically independent and one can also check if their distinctionvia the chosen decomposition corresponds to their separation in height. Again, onecan proceed either using the Shh elements of the matrices of the decomposition or theirunitary polarization vectors.

Firstly, we refer to the sphere and diplane-like matrices because they are the termsgiving the strongest backscattered power contribution and because the scattering mech-anisms they represent are more easily associated to real targets in the scene. The in-terferometric phases relative to the Shh elements of the two terms are represented inFigures 5.12 (a) and (b), whereas in (c) their difference is shown. Looking at the im-ages, one sees that the phase difference is not applicable in a straightforward way forevaluating tree heights: the areas covered with woods do not show any clear variationin this quantity. Though some part of the forests do present interferometric phase dif-ferences between the two mechanisms, they cannot be identified yet as homogeneousand well-defined areas. Another aspect to be considered is the position of the phasecentres corresponding to the two mechanisms. Even when they are correctly identifiedand separated, the question as to where they are placed with respect to the real treesremains. Single bounce phenomena, such as those expressed via the sphere-like ma-trix, can occur not only on the ground but sometimes on the trunk of the imaged tree.Uncertainties are also bound to double bounces; for these, in particular, it is not possi-ble to predict where the scattering centres will be even when related to a ground-trunkdouble bounce, that is, almost the “optimal” scenario. Moreover, the decorrelation dueto random volume, being polarization independent, acts in the same way for the twomechanisms and its influence cannot be used as a further “discriminator” for the phasecentres [PC01].

Similar results were obtained using the sphere and helix terms, leading to the sameconclusions for this other pair of orthogonal mechanisms.

Some more information was obtained from the unitary polarization vectors (see Fig-ure 5.13); in particular, we report here the example of the sphere and the helix terms.The use of these quantities permits us to estimate some phase differences in the areaof the forest as well as on the runway of the airfield. One may anticipate the hy-pothesis of a separation of the scattering centres: the sphere-like mechanism (singlebounce) present mainly on the ground and the depolarization effects, described by thehelix term, basically related to the canopies and hence placed at a variable height be-low the top of the trees. A controversial point is then the interpretation of the samephase difference on the runway, since no separation in height of the scattering centrescould have originated it in this case. An explanation may be found in the differentamount of noise of the two contributions: as put in evidence also by the coherenceimages previously analyzed, a different “partition” of the noise between the terms of


(a) (b)

(c)�� &� �� ' � �� &. �� 0� # ��!��(�� ! � ��

Shh�� . ��' � � �� 0. � � %�� "�

Shh�� . ��' � � �� !�'�� 0.�� # �� '"# � �� (� � � ' � �� &. ��$�(� # ��"!�� !�

� ��%�!��(�� '�� . �(�

the decomposition may be supposed, leading also to separated phase centres (or phasedisplacements between the mechanisms).

Due to the substantial similarity of the Krogager and the Pauli decompositions, thestudy of the phase differences between the terms of this other TD method does notlead to new significant results (see Figure 5.14). Again, the areas where more con-


(a) (b)

(c)� � �� ' � �� . �� 0� #-��!��(�� ! � ��"�/��'�� !�� !��(� � !� � �&'*��# � �&� � � �� ' �$� '�� 0. ��%�� ' � � !�� "� � !��(� � !��$� �&' ��# � �&� � ��(��' �$� '"� �� (.�� # �� '"#$� �� ' � �� . �� (� # ��"!�� /!�� %�!�� '�� . �(�

sistent phase differences may be observed are the runway and the forest. Hence, theconsiderations made regarding the SDH terms can be repeated.

Also the results relative to the Cameron decomposition terms and to the optimalpolarizations give rise to an interferometric phase difference on the airfield runwayand in the forest, as can be seen in Figures 5.15 and 5.16. For the runway, the phase


(a) (b)

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difference cannot be interpreted in terms of a difference in the height of the scatterers.Hence, it seems that it represents only a polarimetric (not a geometric) property of theobserved scatterers.

The phase difference measured in the forested areas, though representing heightvariations of the phase centres, is affected by “purely” polarimetric properties as welland, in fact, the runway results may be considered a confirmation of this fact. Due


(a) (b)

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�� '�!�' � � �"./. ��$�(� # � ��(. � � %�� (#$!�� (� '�� # � �� -�� !�. ��' � � !�� &.�� '�!�' � ��.�. �� 0� #� ��0.�� # � � � � � �� '"# � �� "� �(� � � ' � �� &. �� 0� # ��!��(�� /!�� %�!��(�� '��& � .��

to these considerations, again, the observed interferometric phase differences cannotbe used in a straightforward manner to discriminate the phase centres in vegetatedareas and it stresses the limits of the direct application of decomposition (coherent orincoherent) methods for tree height estimation.

In summary, our analysis does not reveal structured phase differences to be phys-


(a) (b)

(c)�� &� �� ' � �� &. �� 0� # ��!��(��-�� ! � �� &� �$� . !��"��!�� %�� '� �&� �$� . !�� !�� # � � � � � ��'"#$� �� ' � �� &. ��$�(� # ��"!�� /!�� %"!�� '��& �*.��(�

ically meaningful. That is, the role played by geometric considerations seems to belimited. Nevertheless, it puts in evidence other effects, mainly due to the polarimetricproperties of the scatterers, which may be the subject of further, specifically dedicated,investigations.

5.5 - Final considerations 105

5.5 Final considerations

In this chapter we described the interferometric properties of polarizations derivedafter applying TD methods, namely the Krogager decomposition, the Pauli one andthat of Cameron as coherent methods and the optimal polarizations as example ofincoherent methods.

Among the coherent theorems, one should note the use, for the very first time, ofthe Cameron decomposition terms for interferometry. As demonstrated, the interfero-metric coherence of its two components presents aspects which are quite original; noother technique leads to such a separation of the coherence values among the variousterms: the correlation of the max sym term is higher than any of the other decompo-sitions terms considered, while it is also notable that the low backscattered intensityleads the min sym component to have a very low coherence. Moreover, this enhancedseparation characterizes the data as a whole, regardless of the particular homogeneoussubsets of land coverages.

Two topics in particular were investigated: the capability of these methods to re-solve volume decorrelation and to find the exact position of the scattering centres inorder to estimate the height of distributed scatterers; the distinction between volumescattering due to random volumes (that is, vegetated areas) and coherent targets withextension in height (mainly buildings and other man-made artifacts).

We have seen that the scattering mechanisms associated with the various TD methods presentdifferent correlation properties and that, in general, these are strongly influenced by the actualamount of backscattered signal attributed to the given mechanism after the decomposition, aswell as by its SNR [AC01], [ACK02].

As already known from the literature, at least for the optimal polarizations, the dis-tinction among the scattering mechanisms does not allow a precise separation in heightof the scattering centres in the presence of random volumes [PC01]. Hence, an estima-tion of tree height is not possible via a direct measure of the interferometric phase dif-ferences among scattering mechanisms. We verified this for the optimal polarizationsand also extended the analysis to the coherent cases obtaining similar results.

For the Krogager and the Pauli decompositions, the study of the phase differencesbetween the terms did not lead to significant results. The areas where more consis-tent phase differences may be observed are the forest and the runway. Also the resultsrelative to the Cameron decomposition terms, though more evident, give rise to an in-terferometric phase difference mainly in the forest and on the airfield runway. In thefirst case, the characteristics of random volume scattering do not allow the separationof the phase centres and the estimation of the volume extension. For the runway, thephase difference cannot be interpreted in terms of a difference in the height of the scat-terers. Hence, it seems that it represents only a polarimetric (not a geometric) propertyof the observed scatterers.

These investigations do not reveal structured phase differences to be physicallymeaningful. These considerations arise a hitherto important question posed to us bythe scientific community before the study was undertaken, namely, how the polari-metric properties of the decomposition terms are related to the interferometric phasedifferences between them. A subject to be, in our opinion, further investigated.


The results concerning the distinction between random volume scattering and thenewly defined coherent volume scattering were more promising [ACP02a], [AC03]. Foreach mechanism of the various methods, there was evidence of different behaviour of the coher-ence in forests and towns; more precisely, the decompositions confirmed that different sourcesof volume decorrelation exist and we explained why this can be better resolved when it is dueto targets resembling ideal point scatterers. Moreover, we showed that coherent and incoherentTD methods provide similar results in the presence of coherent volume scattering and that theirbehaviour differs increasingly with random volume scattering. This property could be usedas a further source of information in classification processes and should also be takeninto account when estimating the height of coherent extended targets (like buildings), since thetheoretical limits to height estimation of random volumes are no longer valid.

6 Conclusions

This work started from the observation that a variety of representations of polarimetricdata exist and that, in the scientific community, a sort of “competition” is more orless evidently present among them. This situation depends, in part, on the fact thatdifferent polarimetric representations have been developed for pursuing specific tasksand have hence been “ad hoc” fitted to the proposed goal. Thus we decided that ratherthan adopting just one method for a precise scope, that we should consider them alland compare them in the most extensive way. This systematic approach was intendedto verify if substantial differences exist among the various polarimetric observables interms of amount of information they can provide.

The research was then divided into two main parts: a direct analysis of the mainparameters derived from polarimetric SAR data, and a study on those quantities thatare also suitable to interferometric applications.

In Chapter 4, the first part of this study was reported. We compared a set of polari-metric quantities by means of measures of classification accuracy. A first observationwas that little improvement of the classification tests is obtained using some of the more “re-fined” polarimetric parameters derived from the original data rather than using directly theseones [PAC+02], [ACP02b]. Only the ratios of the [S] matrix elements provide resultssignificantly worse than the other polarimetric parameters [AC00]. Another importantconclusion is related to the comparison among single-pixel and averaged-area clas-sifications; we showed that, by averaging the values of the image pixels, the classificationefficiency is, in general, enhanced. This could mean that the parameters under study referto physical properties “spread” over neighbor pixels and correlated. Indeed, the pres-ence of “extended” parameter characteristics may have led to improved classificationaccuracy for increasing sizes of the averaging windows. This is partly in contradictionto the theoretical expectation that some quantities should be related only to point-likescatterers.

In general, better classification results are normally available with optical imagessince the chosen classification algorithms were originally developed for this kind ofdata. Hence, although valuable results could still be obtained with some of the parame-ters under study, the use of classification techniques specifically suited for polarimetricSAR data, like the unsupervised segmentation proposed in [CP97] for the incoherentparameters, would be more appropriate.

The second part of our research dealt with the interferometric applications of someof the polarimetric parameters considered. Chapter 5 described the theoretical aspectsof the subject and the results of our investigations that again consisted mainly of acomparison between coherent and incoherent parameters.

107

108 Chapter 6 - Conclusions

When deriving the interferometric coherence images, we saw that the polarimetricparameters studied give different results and can provide original insights into the degree ofcoherence of the various scattering mechanisms present in a resolution cell [AC01], [ACK02].

A topic of particular interest was the study of the effects of volume decorrelationon the interferometric images. In this context, we introduced the concept of coherentvolume scattering [ACP02a], [AC03] in contrast with the random volume one, for thecase when only varying target heights affects the coherence but the elementary scat-terers are stationary and not randomly distributed. This is the case, for example, forwalls, buildings and other man-made artifacts. An important point is the identifica-tion of the scattering mechanisms present in each resolution cell. This point relatesthe acquired data directly to the physics of the scattering interactions and immediatelyprovides information about the observed scatterers. Our observations seem to confirmthe capability of coherent decomposition theorems in distinguishing volume decorrelation due torandom volumes (that is, tree foliage) from that which is due to height distribution of coherentscatterers (as with man-made artifacts).

A part of Chapter 5 was dedicated to the analysis of the phase difference betweeninterferograms of different decomposition terms. The capability of the target decom-position theorems to distinguish phase centres associated with separated targets wasthen studied referring, in particular, to height estimation of trees and buildings. Thistopic was only briefly addressed and studied in this work, but we were able to con-firm the theoretical expectation that it is not possible to discriminate the phase centres invegetated areas since decorrelation due to random volume is polarization independent [PC01].Hence, also with coherent TDs, tree height determination is not possible by a straight-forward inversion of the difference in interferometric phases. On the contrary, the useof these techniques for coherent volume scattering resolution and height estimation ismore promising and, therefore, suggested as a topic for further investigation.

A Relationships among polarizationgeometrical parameters

Let us show how Equations (2.31) and (2.32) are derived in [BW85], in the most generalcase of elliptic polarization and starting from an EM wave written as (2.23), i. e.:

~E = (E2h+ E2

v)1/2[cosα ejδhh+ sinα ejδv v] · exp j(ωt− ~k · ~r). (A.1)

Writing for simplicity:

a1 = (E2h+ E2

v)1/2 cosα, (A.2)

a2 = (E2h+ E2

v)1/2 sinα (A.3)

andτ = ωt− ~k · ~r, (A.4)

one can derive the real part of the components of ~E as:{

Eh = a1 cos(τ + δh)Ev = a2 cos(τ + δv).

(A.5)

In general, the axes of the ellipse are not in the h and v directions. Let us reconsiderFigure 2.3 and let (0, x, y) be a new reference system with the axes along the axes of theellipse so that ψ (0 ≤ ψ < π) is the angle between h and the direction x of the majoraxis. Then the components Ex and Ey are related to Eh and Ev by:

{

Ex = Eh cosψ + Ev sinψEy = −Eh sinψ + Ev cosψ.

(A.6)

If 2a and 2b (a ≥ b) are the lengths of the axes of the ellipse, the equation of theellipse in the new reference system is:

{

Ex = a cos(τ + δ0)Ey = ±b sin(τ + δ0),

(A.7)

indicating that the relative phase δ of the two orthogonal components is now expressedby:

δ = π/2 + δ0. (A.8)

The two signs in (A.7) distinguish the two possible directions in which the end pointof the electric vector may describe the ellipse.

109

110 Appendix A - Relationships among polarization geometrical parameters

x

h

b

v

χ

ψa

y

�� +-�� !��0� � !�� &' �� "�(��

To determine a and b we compare (A.6) and (A.7) and use (A.5):

a(cos τ cos δ0 − sin τ sin δ0) == a1 cos(τ + δh) cosψ + a2 cos(τ + δv) sinψ == a1(cos τ cos δh − sin τ sin δh) cosψ + a2(cos τ cos δv − sin τ sin δv) sinψ

±b(sin τ cos δ0 + cos τ sin δ0) == −a1 cos(τ + δh) sinψ + a2 cos(τ + δv) cosψ == −a1(cos τ cos δh − sin τ sin δh) sinψ + a2(cos τ cos δv − sin τ sin δv) cosψ.

(A.9)

Next we equate the coefficients of cos τ and sin τ :

a cos δ0 = a1 cos δh cosψ + a2 cos δv sinψ (A.10)a sin δ0 = a1 sin δh cosψ + a2 sin δv sinψ (A.11)±b cos δ0 = a1 sin δh sinψ − a2 sin δv cosψ (A.12)±b sin δ0 = −a1 cos δh sinψ + a2 cos δv cosψ. (A.13)

By squaring and adding the two terms with a and the two terms with b we obtain:

a2 = a21 cos2 ψ + a2

2 sin2 ψ + 2a1a2 cosψ sinψ cos δ (A.14)b2 = a2

1 sin2 ψ + a22 cos2 ψ − 2a1a2 cosψ sinψ cos δ. (A.15)

Hence:a2 + b2 = a2

1 + a22. (A.16)

Appendix A - Relationships among polarization geometrical parameters 111

Next we multiply (A.10) by (A.12), (A.11) by (A.13) and add. This gives:

∓ab = a1a2 sin δ. (A.17)

Further on, dividing (A.12) by (A.10) and (A.13) by (A.11) we obtain:

± b

a=a1 sin δh sinψ − a2 sin δv cosψ

a1 cos δh cosψ + a2 cos δv sinψ=

=−a1 cos δh sinψ + a2 cos δv cosψ

a1 sin δh cosψ + a2 sin δv sinψ, (A.18)

and, leaving aside ±b/a, multiplying both the remaining terms by the product of thetwo denominators and adding and subtracting the angles, these relations give the fol-lowing equation for ψ:

(a21 − a2

2) sin 2ψ = 2a1a2 cos δ cos 2ψ. (A.19)

Considering now the angle α (0 ≤ α ≤ π/2), such that:

a2

a1

= tanα, (A.20)

Equation (A.19) then becomes:

tan 2ψ =2a1a2

a21 − a2

2

cos δ =2 tanα

1 − tan2αcos δ, (A.21)

i. e.:tan 2ψ = tan 2α cos δ. (A.22)

Now, from (A.16) and (A.17) we also have:

∓ 2ab

a2 + b2=

2a1a2

a21 + a2

2

sin δ =2 tanα

1 + tan2αsin δ = sin 2α sin δ. (A.23)

Let χ (−π/4 ≤ χ ≤ +π/4) be another auxiliary angle, such that:

tanχ = ∓ b

a. (A.24)

The numerical value of tanχ represents the ratio of the axes of the ellipse and the signof χ distinguishes the two directions in which the ellipse may be described. Equa-tion (A.23) may then be rewritten in the form:

sin 2χ = sin 2α sin δ. (A.25)

B Target decomposition theorems

B.1 Krogager decomposition

The following details about the Krogager decomposition method are taken from [Kro93]and [KC95].

Due to reciprocity, in monostatic configurations, the matrix [S] has the form:

[S] =

[

a+ b cc a− b

]

(B.1)

(see the general expression (2.120)).

According to Krogager, after mathematical manipulations of the complex elementsa, b and c, [S] can be decomposed as shown in (2.121):

[S] = ks[S]sphere + ej(φb−φa)(kd[S]diplane + kh[S]helix), (B.2)

assuming phase-normalization with respect to a.

ks, kd and kh are real quantities whose values are given by:

ks = |a| , (B.3)

kd = |b|√

(1 − |Im{d}|)2 + (Re{d})2 , (B.4)kh = 2|Im{d}||b| , (B.5)

where:d =

c

b, (B.6)

while the matrices expressing scattering from a sphere, a diplane and a right- or left-wound helix have the form:

[S]sphere =

[

1 00 1

]

, (B.7)

[S]diplane =

[

cos 2θ sin 2θsin 2θ − cos 2θ

]

, (B.8)

with:tan 2θ =

Re{d}1 − |Im{d}| , (B.9)

and

[S]helix =1

2

[

1 ±j±j −1

]

. (B.10)

113

114 Appendix B - Target decomposition theorems

The sense of the helix component (the sign of j) is chosen according to sign(φd) =sign(φc − φb).

However, this representation does not have the desired roll-invariant property, whichmeans that the parameters calculated in this way are not all independent of the orien-tation angle around the line of sight.

To avoid this problem, one can adopt a slightly different model, namely [KC95]:

[S] = ejφ{ejφsks[S]sphere + kd[S]diplane(θ) + kh[S]helix(θ)} =

= ejφ

{

ejφsks

[

1 00 1

]

+ kd

[

cos 2θ sin 2θsin 2θ − cos 2θ

]

+ khe∓j2θ

2

[

1 ±j±j −1

]}

,

(B.11)

where both the diplane and the helix are considered rotated by an angle θ, and expressit (considering only a right-wound helix) in terms of the right-left circular polarizationbasis:

[SC] = ejφ

{

ejφsks

[

0 jj 0

]

+ kd

[

ej2θ 00 −e−j2θ

]

+ kh

[

ej2θ 00 0

]}

=

= ejφ

[

(kd + kh)ej2θ jkse

jφs

jksejφs −kde

−j2θ

]

. (B.12)

Let us now consider a general symmetric (i. e., having SRL = SLR) scattering matrixin this basis:

[SC] =

[

SRR SRL

SRL SLL

]

=

[

|SRR|ejφRR |SRL|ejφRL

|SRL|ejφRL −|SLL|ej(φLL−π)

]

(B.13)

and define:

φ1 = φRR + φLL − π , (B.14)φ2 = φRR − φLL + π , (B.15)

which inserted in (B.13) yield:

[SC] = ej 1

2φ1

[

|SRR|ej 1

2φ2 |SRL|ej(φRL− 1

2φ1)

|SRL|ej(φRL− 1

2φ1) −|SLL|e−j 1

2φ2

]

. (B.16)

Now, if |SRR| > |SLL|, we may write:

[SC] = ej 1

2φ1

[

[(|SRR| − |SLL|) + |SLL|]ej 1

2φ2 |SRL|ej(φRL− 1

2φ1)

|SRL|ej(φRL− 1

2φ1) −|SLL|e−j 1

2φ2

]

, (B.17)

that, compared with (B.12), yields:

ks = |SRL|, (B.18)k+

d = |SLL|, (B.19)k+

h = |SRR| − |SLL|. (B.20)

B.2 - Cameron decomposition 115

Similarly, if |SLL| > |SRR|, one can compare (B.17) with the analogue of (B.12) writtenfor a left helix and derive:

k−d = |SRR|, (B.21)k−h = |SLL| − |SRR|. (B.22)

The angles defined in (B.12) can be obtained as well, by comparison, as:

φ =1

2(φRR + φLL − π), (B.23)

θ =1

4(φRR − φLL + π), (B.24)

φs = φRL − 1

2(φRR + φLL). (B.25)

All the parameters so calculated have the desired orientation independence, so that,in general, one has to simply perform a change of basis to RL-circular in order to derivethem.

B.2 Cameron decomposition

The decomposition method proposed by Cameron [CL92], [CYL96] is based on therepeated extraction of the basic properties of the scatterers, like reciprocity and sym-metry, by means of projections of the measured scattering vectors onto the relativesubspaces or subsets of reciprocal, symmetric, etc., scattering vectors. Indeed, in thegeneral case of bistatic configurations, the reciprocity constraint is not given and themost basic property possessed by a scatterer which can be derived from its [S] matrixis just its tendency to be approximately reciprocal. Moreover, even in the monostaticcase reciprocity is sometimes violated, due to propagation effects or interactions withspecial materials (for example, materials whose interaction with the EM field is non-linear).

When considering the space of the scattering vectors (expressing them in the lexi-cographic basis so that k(4)L represents [S]), those corresponding to reciprocal targetsoccupy the subspace Vrec ⊂ C

4 generated by the projection matrix [Prec],

[Prec] =

1 0 0 00 1/2 1/2 00 1/2 1/2 00 0 0 1

. (B.26)

Any scattering matrix can be uniquely decomposed into two components orthogonalin C

4, krec and knon−rec, with: krec ∈ Vrec and knon−rec ∈ Vnon−rec (Vnon−rec is the subspaceof C

4 which is orthogonal to Vrec).

The decomposition is then given by:

k(4)L = krec + knon−rec , (B.27)

116 Appendix B - Target decomposition theorems

with:

krec = [Prec]k(4)L , (B.28)knon−rec = ([I] − [Prec])k(4)L . (B.29)

The degree to which a scattering matrix obeys reciprocity can be measured by compar-ing the magnitude of the two components, krec and knon−rec. This measure is given bythe parameter θrec, which is the angle between the scattering vector and the subspaceVrec of the vectors corresponding to reciprocal scattering matrices:

θrec = arccos∥

∥

∥[Prec]k(4)L

∥

∥

∥, 0 ≤ θrec ≤

π

2, (B.30)

where:k(4)L =

k(4)L

‖k(4)L‖. (B.31)

Scattering matrices with θrec = 0 correspond to scatterers which strictly obey the reci-procity principle whereas scattering matrices with θrec = π/2 lie entirely in the com-plement of Vrec and thus violate the reciprocity principle.

A reciprocal scattering matrix can be further decomposed into components relatedto most dominant symmetric and least dominant symmetric scatterers (intended hereas scatterers having an axis of symmetry in the plane orthogonal to the radar line-of-sight). This division, unfortunately, does not define two orthogonal subspaces of Vrec,since both components belong to the subset Wsym of the symmetric scattering vectors,which is simply a subset of Vrec.

To effect this separation, one uses the three reciprocal matrices (normalized) of thePauli set:

[Sa] =1√2

[

1 00 1

]

, (B.32)

[Sb] =1√2

[

1 00 −1

]

, (B.33)

[Sc] =1√2

[

0 11 0

]

(B.34)

and their corresponding scattering vectors ka, kb and kc. Since for a generic krec ∈ Vrec

it holds:krec = αka + βkb + γkc , (B.35)

being α, β, γ ∈ C, it is then possible to write:

krec = [Psym]krec + ([I] − [Psym])krec = kmaxsym + kmin

sym, (B.36)

defining [Psym] as:

kmaxsym = [Psym]krec = 〈krec|ka〉ka + 〈krec|k′〉k′, (B.37)

with:k′ = cos θ kb + sin θ kc . (B.38)

B.2 - Cameron decomposition 117

In the equation above, θ is chosen such that |〈krec|k′〉| is a maximum. Within thesehypotheses, it is guaranteed that kmax

sym is the largest symmetric component that can beextracted from krec. As anticipated, both the most dominant and the least dominantsymmetric components belong to Wsym and it can be proven that they are orthogonal.

[Psym] is defined by introducing k′ given by (B.38), with θ satisfying:

θ =1

2χ , (B.39)

where:sinχ =

βγ∗ + β∗γ√

(βγ∗ + β∗γ)2 + (|β|2 − |γ|2)2

(B.40)

and

cosχ =|β|2 − |γ|2

√

(βγ∗ + β∗γ)2 + (|β|2 − |γ|2)2

. (B.41)

Again, the degree to which krec deviates from belonging to the subset Wsym of thesymmetric scattering vectors is measured by the angle τ :

τ = arccos

∣

∣

∣

∣

〈krec|[Psym]krec〉‖krec‖ · ‖[Psym]krec‖

∣

∣

∣

∣

, 0 ≤ τ ≤ π

4. (B.42)

If τ = 0 then krec ∈ Wsym. The maximum asymmetry condition, τ = π4

occurs, on thecontrary, if and only if krec is the scattering vector of a left or right helix.

Equation (B.30) may be generalized in order to compare an arbitrary scattering ma-trix [S] with a given test one [Stest]. As reported in [CYL96], it is in fact possible todefine a metric which compares normalized scattering matrices belonging to Wsym.Such a metric is independent of the overall relative phases of the scattering matricesand independent of the orientations (rotations about the line-of-sight axis) of the scat-terers represented. In this way, one can iteratively split a scattering matrix into severalcomponents by evaluating its properties (reciprocity, symmetry) and comparing it withtypical ones (such as helices, diplanes, dipoles, etc.)

C Classification results

The classification tests discussed in Chapter 4 lead to numerical estimates of their ac-curacy expressed in different forms (confusion matrices, error measures, etc.). For sakeof compactness, we report here only the main results of those performed on the dataset of October ‘99 in terms of producer’s and user’s accuracy and of omission andcommission errors.

We choose for the tests on the coherent parameters the measures relative to thesingle-pixel basis and to the largest window. For those on the incoherent parameters(entropy, α and anisotropy), the measures concerning the 3×3-pixel averaging windowhave been given as smallest size case.

119

120 Appendix C - Classification results

C.1 Backscattered wave amplitude

[S] matrix elements (single-pixel basis) - Max. Lik. classification test:

• Overall accuracy = (62243/150262) 41.42%

• Kappa coefficient = 0.316

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Class Pr. Acc. (%) Us. Acc. (%) Pr. Acc. (pixels) Us. Acc. (pixels)water 21.34 29.75 4774/22374 4774/16048

houses 28.71 84.87 6744/23492 6744/7946roads 78.70 34.89 15411/19582 15411/44170trees 76.07 57.72 17164/22562 17164/29737grass 10.54 22.75 1875/17784 1875/8242field 1 45.28 43.46 11942/26372 11942/27476field 2 23.94 26.03 4333/18096 4333/16643

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�� %�!��

Class Comm. (%) Om. (%) Comm. (pixels) Om. (pixels)water 70.25 78.66 11274/22374 17600/22374


C.1 - Backscattered wave amplitude 121

[S] matrix elements (single-pixel basis) - Min. Dist. classification test:



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C.2 Characteristic polarizations

[S] matrix co- and cross-polar nulls (single-pixel basis) - Max. Lik. classification test:


• Kappa coefficient = 0

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C.2 - Characteristic polarizations 127

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C.3 Parameters of the coherent target decomposition the-orems

SDH decomposition terms (single-pixel basis) - Max. Lik. classification test:



��!�% � � �� + # # � !�' � � � + # # � �� . !�� !��$� � � � � �� /! �� # � ! �(� � ��# !�� &' � �� &�(. � /��' �� )�� # �&.�� $� ��' � ��0. � � � � '��

�� %�!��



��!�% � � �&� � �� &.�./� �0� � �&' !�'� ��.�� 0� � �&' ��(� �&�"�� $� . !� ��&� �� !� � �� "� �/! �"� �� #� !��(� � � #$!��$� �&'� �� &�0. � ��&' �� )�� #$�&.�� $� ��' � ��0.*� � � � '"� � �

�� %�!��



C.3 - Parameters of the coherent target decomposition theorems 133

SDH decomposition terms (single-pixel basis) - Min. Dist. classification test:



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�� %�!��



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�� %�! � � � �(�




SDH decomposition terms (single-pixel basis) - Parall. classification test:

• Overall accuracy = (39074/150262) 26%


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�� %�!��



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�� %�!��




SDH decomposition terms (15×15-pixel averaging window) - Max. Lik. classificationtest:



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15×15 �� ! �� ! �&� '�� /� '� ��



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15×15 �� ! �� ! �&� '�� /� '� ��




SDH decomposition terms (15×15-pixel averaging window) - Min. Dist. classifica-tion test:



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15×15 �� ! � ��! �� '�� /� '� �� (�



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15×15 �� ! �� ! �&� '�� /� '� ��




SDH decomposition terms (15×15-pixel averaging window) - Parall. classificationtest:



� !�% � � �&� � � � � ��+ # #��!�'� � � ��+ # #��&� � �$� . !�� !�� "� ��!�� !�� # � ! �(� � ��# !�� &'�� &�0. � ��&' �� )�� #$�&.�� &' � ��0.*� �

15×15 �� ! �� ! �&� '�� /� '� ��



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15×15 �� ! �� ! �&� '�� /� '� ��




Pauli decomposition terms (single-pixel averaging window) - Max. Lik. classifica-tion test:



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�� %�! � � � �(�



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�� %�! � � � �(�




Pauli decomposition terms (single-pixel averaging window) - Min. Dist. classifica-tion test:



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�� %�! � � � �(�



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�� %"!�� (�




Pauli decomposition terms (single-pixel averaging window) - Parall. classificationtest:



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�� %�! � � � �(�



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�� %�! � � � �(�




Pauli decomposition terms (15×15-pixel averaging window) - Max. Lik. classificationtest:



� !�% � � �&� � �� + # # � !�'� � � � + # # � �� . !�� !��$� � � � � �� /! �"� �� # � ! �(� � ��# !�� &' � �� &�0. � ��&' �� !�� #$��./�"� � � � � �&' � ��(.*� �

15×15 �� ! �� ! �&� '�� /� '� �� (�



� !�% � � �&� � � � ��&.�.�� 0� � �&'-!�'� �&.�� 0� � �&'-��(� �&�"�� . !� ��&� �� !�� "� �/! �"� �� #� !��0� � ��#$!��$� ��'� �� &�0. � ��&' �� !�� # �&.�� $� �&' � ��(.*� �

15×15 �� ! ��! �� '�� /� '� �� (�




Pauli decomposition terms (15×15-pixel averaging window) - Min. Dist. classifica-tion test:



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15×15 �� ! �� ! �&� '�� /� ' ��



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15×15 �� ! ��! �&� '�� /� '� �� (�




Pauli decomposition terms (15×15-pixel averaging window) - Parall. classificationtest:



� !�% � � �&� � �� + # #��!�'� � � ��+ # #��&� � �$� . !�� !�� "� ��!�� !�� # � ! �(� � ��# !�� &'�� &�0. � ��&' �� !�� #$��./�"� � � � � �&' � ��(.*� �

15×15 �� ! �� ! �&� '�� /� '� �� (�



� !�% � � �&� � � � � �&.�./� �0� � �&' !�'� ��&.�� 0� � �&'/��(� �&� �� . !� �� !�� !��!�� #� !��0� � ��#$!��$� ��'� �� &�0. � ��&' �� !�� # �&.�� $� �&' � ��(.*� �

15×15 �� ! ��! �� '�� /� '� �� (�




Cameron decomposition terms (single-pixel averaging window) - Max. Lik. classifi-cation test:



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�� %�!�� (�



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�� %�!��




Cameron decomposition terms (single-pixel averaging window) - Min. Dist. classi-fication test:

• Overall accuracy = 56502/150262) 37.60%


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�� %�!�� (�



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�� %�!��




Cameron decomposition terms (single-pixel averaging window) - Parall. classifica-tion test:



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�� %�!�� (�



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�� %�!��




Cameron decomposition terms (15×15-pixel averaging window) - Max. Lik. classi-fication test:



� !�% � � �&� � � � ��+ # # � !�'� � � � + # # � �� . !�� !��$� � � � � �� /! �"� �� # � ! �(� � ��# !�� &' � �� &�0. � ��&' �� !�. ��&' ��# �&.�� $� ��' � ��0.*� �

15×15 �� ! � ��!��&� '"� � � '� ��"� � �



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15×15 �� ! �� ! �&� '�� /� '� ��




Cameron decomposition terms (15×15-pixel averaging window) - Min. Dist. classi-fication test:



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15×15 �� ! ��! �� '�� /� '� �� (�



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15×15 �� ! ��! �&� '�� /� '

� ��




Cameron decomposition terms (15×15-pixel averaging window) - Parall. classifica-tion test:



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15×15 �� ! � ��!��&� '"� � � '� ��"� � �



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15×15 �� ! �� ! �&� '�� /� '� ��




C.4 Entropy/α parameters

H and α (3×3-pixel averaging window) - Max. Lik. classification test:



��!�% � � �� + # # � !�' � � � + # # � �� . !�� !��$� � � � � �� /! �� # � ! �(� � ��# !�� &' � �� &�(. � /��' ��

H!�'�

α��!�� !�. ��

3×3 �� ! ��! �&� '�� /� '� �� (�



��!�% � � �&� � � � � �&.�./� �0� � �&' !�'� ��.�� 0� � �&' ��(� �&�"�� $� . !� ��&� �� !� � �� "� �/! �"� �� #� !��(� � � #$!��$� �&'� �� &�0. � ��&' ��

H!�'�

α��!��!�. ��

3×3 �� ! ��! �� '�� /� '� �� (�



C.4 - Entropy/α parameters 151

H and α (3×3-pixel averaging window) - Min. Dist. classification test:



� !�% � � �&� � �� &+ # # ��!�'� � � �&+ # # �� $� . !�� !��$� � � � � ��"� �� '�� #� !��0� � ��#$!� � �&' � �� &�0. � /��' ��

H!�'�

α��!�� !�. ��

3×3 �� ! ��! �� '�� /� '� �� (�



� !�% � � �&� �� .�./� �(� � ��'/!�'� ��&.�� (� � �&'/��(� �&� �� . !�� !��$� � � � � �� '�� #� !��0� � ��#$!�� &' � �� &�0. � ��&' ��"�

H!�'�

α�"!��!�. ��

3×3 �� ! � ��!��&� '"� � � '� ��"� � �




H and α (3×3-pixel averaging window) - Parall. classification test:



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H!�'�

α��!�� !�. ��

3×3 �� ! ��! �&� '�� /� '� �� (�



��!�% � � �&� � �� ./.�� (� � ��' !�'� ��./� �(� � ��' ��(� �&� �� $� . !�� !� � �� "� � !��!�� #� !��(� � � #$!��$� �&'� �� &�0. � ��&' ��

H!�'�

α��!��!�. ��

3×3 �� ! ��! �� '�� /� '� �� (�




H and α (15×15-pixel averaging window) - Max. Lik. classification test:



� !�% � � �&� �� + # # � !�'� � � � + # # � �� . !�� !��$� � � � � �� /! �"� �� # � ! �(� � ��# !�� &' � �� &�0. � ��&' ��

H!�'�

α��!��!�. ��

15×15 �� ! � ��!��&� '"� � � '� ��"� � �



� !�% � � �&� �� &.�.�� 0� � �&'-!�'� �&.�� 0� � �&'-��(� �&�"�� . !� ��&� �� !�� "� �/! �"� �� #� !��0� � ��#$!��$� ��'� �� &�0. � ��&' ��

H!�'�

α�"!��!�. ��

15×15 �� ! �� ! �&� '�� /� '� ��




H and α (15×15-pixel averaging window) - Min. Dist. classification test:



��!�% � � �&� �� +�# #�� !�'� � � �&+ # #�� . !�� !�� ' � � � � �$� #� !��0� � ��#$!��$� ��' � �� "�� &�(. � ��&' ��

H!�'�

α��!�� !�. ��

15×15 �� ! ��! �� '�� /� '� �� (�



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H!�'�

α��!��!�. ��

15×15 �� ! �� ! �&� '�� /� '� ��




H and α (15×15-pixel averaging window) - Parall. classification test:



� !�% � � �&� � � � ��+ # #��!�'� � � ��+ # #��&� � �$� . !�� !�� "� ��!�� !�� # � ! �(� � ��# !�� &'�� &�0. � ��&' ��

H!�'�

α��!��!�. ��

15×15 �� ! � ��!��&� '"� � � '� ��"� � �



� !�% � � �&� � � � � �&.�./� �0� � �&' !�'� ��&.�� 0� � �&'/��(� �&� �� . !� �� !�� !��!�� #� !��0� � ��#$!��$� ��'� �� &�0. � ��&' ��

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Short resume of Vito AlbergaPERSONAL DATA

DATE OF BIRTH November, 2nd 1970CITIZENSHIP ItalianOFFICE ADDRESS German Aerospace Centre (DLR) - Microwaves and Radar Insti-

tute, P.O. Box 1116, D-82234 Wessling, GermanyE-MAIL [email protected]

EDUCATION1998 Master’s Degree in Physics (Laurea in Fisica): ”The Use of SAR Data for

Environmental-Risk Areas Monitoring”. University of Bari, Italy1998 Final Master’s Research Work at the Institute of Intelligent Systems for Automa-

tion (ISSIA) of the National Research Council (CNR), Bari, Italy1989 High School Leaving Certificate (Diploma di Maturita Scientifica), Bari, Italy

CONTINUING EDUCATION2002 CCG Seminar “SAR Principles and Applications”; DLR, Oberpfaffenhofen, Ger-

many2001 Training Workshop “SAR Polarimetry and Polarimetric Interferometry”; Uni-

versity of Rennes, France2000 Training Workshop “Polarimetric Radar Remote Sensing: Basic Concepts, The-

ory, Systems and Applications”; DLR, Oberpfaffenhofen, Germany2000 Training Workshop “On the Foundations of Polarimetry and Basic Principles of

Polarimetric Radar-Observations and Analysis”; JRC, Ispra (VA), Italy1999 “IDL Intensive Course”; CREASO GmbH, Gilching bei Munchen, Germany1998 “Seminar on ERS SAR and Other Complementary Spaceborne Sensors for Land

Use and Land Cover Applications”; ESA/ESRIN, Frascati (Roma), Italy1998 CNR/ISIATA Summer School “Instrumental Techniques in Environmental At-

mospheric Physics”; Castro Marina (LE), Italy

WORKING EXPERIENCE2002 TO DATE Performance Engineer for the project TerraSAR-X at the Mi-

crowaves and Radar Institute (HR) of the German AerospaceCentre; Oberpfaffenhofen, Germany

2002 Collaboration with Definiens-Imaging for the implementation ofalgorithms for the analysis of polarimetric SAR data into theEcognition software

1999 - 2002 EU Fellow at DLR/HR. Research activity: studies on polarimetryand polarimetric SAR interferometry. Teaching activity: tutoringof students at Master’s Degree level

PAST EXPERIENCES Summer jobs abroad, private lessons of mathematics.

171

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